FA CUL TY L inearalgebraicgroupstdemedts/files/lag.pdf · groups over any field, in particular the...
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LINEAR ALGEBRAIC GROUPS Tom De Medts Department of Mathematics: Algebra and Geometry Master of Mathematics Academic year 2019–2020 The picture on this front page is taken from “The Book of Involutions”. It represents a Dynkin diagram of type D4.
Transcript of FA CUL TY L inearalgebraicgroupstdemedts/files/lag.pdf · groups over any field, in particular the...
LINEAR ALGEBRAIC GROUPS Tom De Medts Department of Mathematics: Algebra and Geometry Master of Mathematics Academic year 2019–2020
The picture on this front page is taken from “The Book of Involutions”. It represents a Dynkin diagram of type D4.
Preface
The leading pioneer in the development of the theory of algebraicgroups was C. Chevalley. Chevalley’s principal reason for interestin algebraic groups was that they establish a synthesis between the twomain parts of group theory — the theory of Lie groups and the the-ory of finite groups. Chevalley classified the simple algebraic groupsover an algebraically closed field and proved the existence of analogousgroups over any field, in particular the finite Chevalley groups.
—R.W. Carter
Linear algebraic groups are matrix groups defined by polynomials; a typi-cal example is the group SLn of matrices of determinant one. The theoryof algebraic groups was inspired by the earlier theory of Lie groups, and theclassification of algebraic groups and the deeper understanding of their struc-ture was one of the important achievements of last century, mainly led byA. Borel, C. Chevalley and J. Tits.
The three main standard references on the topic are the books by Borel[Bor91], Humphreys [Hum75] and Springer [Spr09]. However, they all threehave the disadvantage of taking the “classical” approach to algebraic groups,or more generally to algebraic geometry. Also the recent book by Malle andTesterman [MT11] follows this approach.
We have opted to follow the more modern approach, which describes al-gebraic groups as functors, and describes their coordinate algebra as Hopfalgebras (which are not necessarily reduced, in constrast to the classical ap-proach). This essentially means that we take the scheme-theoretical point ofview on algebraic geometry. This might sound overwhelming and needlesslycomplicated, but it is not, and in fact, we will only need the basics in order todevelop a deep understanding of linear algebraic groups; and it will quicklybecome apparent that this functorial approach is very convenient.
Of course, this approach is not new, and the first reference (which is stillan excellent introduction to the subject) is the book by Waterhouse [Wat79].Some more modern and more expanded versions have been written, andthe current lecture notes are based mainly on the online course notes byMilne [Mil12a, Mil12b, Mil12c], McGerty [McG10] and Szamuely [Sza12]. In
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fact, some paragraphs have been copied almost ad verbatim, and I shouldperhaps apologize for not mentioning these occurences throughout the text.The interested reader who wants to continue reading in this direction, and inparticular wants to understand the theory over arbitrary fields, should havea look at Chapter VI of the Book of Involutions [KMRT98], which is rathercondensely written, but in the very same spirit as the approach that we aretaking in these course notes.
So why another version? For several reasons: I found the course notesof Milne, although extremely detailed and complete, in fact too detailed,and very hard to use in a practical course with limited time. In contrast,McGerty’s notes —which unfortunately seem to have disappeared from theweb— are too condensed for a reader not familiar with the topic. Szamuely’snotes are very much to the point, but they don’t go deep enough into thetheory at various places. Finally, these course notes were written to be usedin a Master course in Ghent University, and they are especially adapted to thebackground knowledge and experience of the students following this course.
This is why these course notes take off with a fairly long preliminarypart: after an introductory chapter, there is a chapter on algebras, a chapteron category theory, and a chapter on algebraic geometry. Linear algebraicgroups —the main objects of study in this course— will be introduced onlyin Chapter 5.
I have chosen the classification of reductive linear algebraic groups overalgebraically closed fields as the ultimate goal in this course. Of course,there is much to do beyond this —in some sense, the interesting things onlystart happening when we leave the world of algebraically closed fields— butalready reaching this point is quite challenging. In particular, and mainly inthe later chapters, some of the proofs have been omitted. I have neverthelesstried to indicate the lines of thought behind the structure theory, and myhope is that a reader who has reached the end of these course notes will haveacquired some feeling for the theory of algebraic groups.
One of the main shortcomings in these course notes is the lack of (more)examples. However, the idea is that these course notes are accompanied byexercise classes, and this is where the examples should play a prominent role.
Tom De Medts (Ghent, January 2013)
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Of course, the original version of these course notes contained severaltypos and other little mistakes, which have now been corrected. There willalmost certainly still be some other mistakes left, and any comments orsuggestions for improvements are appreciated!
Tom De Medts (Ghent, May 2013)
Again, some minor mistakes have been corrected. I have rewritten thebeginning of section 5.4, which now contains a rigorous approach to comod-ules, and section 7.2, where I have simplified the definition of the Lie bracket(at the cost of having to omit the proofs). Needless to say, comments andsuggestions remain welcome.
Tom De Medts (Ghent, September 2014)
A number of typos and minor mistakes remained and have now been elim-inated; hopefully the number of mistakes in these course notes will eventuallyconverge to zero. I have also added a number of additional clarifications andexamples.
Tom De Medts (Ghent, January 2016)
It was time again for some more substantial changes. Section 5.3 is new,and was necessary to deal with some subtleties concerning quotients thatwere not accurately dealt with in the earlier versions. Section 8.5 has alsobeen rewritten (these were two separate sections).
Tom De Medts (Ghent, August 2017)
Milne’s lecture notes are now available as a book that I can highly rec-ommend for further reading [Mil17].
Tom De Medts (Ghent, February 2019)
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Contents
Preface iii
1 Introduction 1
1.1 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The building bricks . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Finite algebraic groups . . . . . . . . . . . . . . . . . . 3
1.2.2 Abelian varieties . . . . . . . . . . . . . . . . . . . . . 3
1.2.3 Semisimple linear algebraic groups . . . . . . . . . . . 3
1.2.4 Groups of multiplicative type and tori . . . . . . . . . 5
1.2.5 Unipotent groups . . . . . . . . . . . . . . . . . . . . . 5
1.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 Solvable groups . . . . . . . . . . . . . . . . . . . . . . 6
1.3.2 Reductive groups . . . . . . . . . . . . . . . . . . . . . 6
1.3.3 Disconnected groups . . . . . . . . . . . . . . . . . . . 7
1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Algebras 9
2.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . 9
2.2 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Tensor products of K-modules . . . . . . . . . . . . . . 12
2.2.2 Tensor products of K-algebras . . . . . . . . . . . . . . 15
3 Categories 19
3.1 Definition and examples . . . . . . . . . . . . . . . . . . . . . 19
3.2 Functors and natural transformations . . . . . . . . . . . . . . 21
3.3 The Yoneda Lemma . . . . . . . . . . . . . . . . . . . . . . . 25
4 Algebraic geometry 31
4.1 Affine varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 The coordinate ring of an affine variety . . . . . . . . . . . . . 36
4.3 Affine varieties as functors . . . . . . . . . . . . . . . . . . . . 40
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5 Linear algebraic groups 43
5.1 Affine algebraic groups . . . . . . . . . . . . . . . . . . . . . . 43
5.2 Closed subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3 Homomorphisms and quotients . . . . . . . . . . . . . . . . . 54
5.4 Affine algebraic groups are linear . . . . . . . . . . . . . . . . 56
6 Jordan decomposition 63
6.1 Jordan decomposition in GL(V ) . . . . . . . . . . . . . . . . . 63
6.2 Jordan decomposition in linear algebraic groups . . . . . . . . 67
7 Lie algebras and linear algebraic groups 73
7.1 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.2 The Lie algebra of a linear algebraic group . . . . . . . . . . . 76
8 Topological aspects 83
8.1 Connected components of matrix groups . . . . . . . . . . . . 83
8.2 The spectrum of a ring . . . . . . . . . . . . . . . . . . . . . . 84
8.3 Separable algebras . . . . . . . . . . . . . . . . . . . . . . . . 87
8.4 Connected components of linear algebraic groups . . . . . . . 90
8.5 Dimension and smoothness . . . . . . . . . . . . . . . . . . . . 94
9 Tori and characters 97
9.1 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
9.2 Diagonalizable groups . . . . . . . . . . . . . . . . . . . . . . . 98
9.3 Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
10 Solvable linear algebraic groups 105
10.1 The derived subgroup of a linear algebraic group . . . . . . . . 105
10.2 The structure of solvable linear algebraic groups . . . . . . . . 107
10.3 Borel subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . 113
11 Semisimple and reductive groups 117
11.1 Semisimple and reductive linear algebraic groups . . . . . . . . 117
11.2 The root datum of a reductive group . . . . . . . . . . . . . . 121
11.3 Classification of the root data . . . . . . . . . . . . . . . . . . 131
References 136
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Chapter 1 Introduction
Before we formally develop the theory of linear algebraic groups, we willgive some examples that should give an impression of what to expect fromthis theory. We will also give an overview of the different types of algebraicgroups that we will (or will not) encounter.
1.1 First examples
Definition 1.1.1. Let k be a (commutative) field. Roughly speaking, analgebraic group over k is a group that is defined by polynomials, by which wemean that the underlying set is defined by a system of polynomial equations,and also that the multiplication and the inverse in the group are given bypolynomials. If the underlying set is defined as a subset of kn (for some n),then we call it an affine algebraic group, and one of the fundamental resultsthat we will prove later actually shows that every affine algebraic group isa linear algebraic group in the sense that it can be represented as a matrixgroup.
This definition admittedly is rather vague; we will later give a much moreprecise definition, which will require quite some more background, so theabove definition will do for now. Some examples will clarify what we have inmind.
Examples 1.1.2. (1) SLn.
If A = (aij) ∈ Matn(k) is an arbitrary matrix, then A belongs to SLn(k)if and only if
detA =∑
σ∈Symn
sgn(σ) · a1,σ(1) · · · an,σ(n) = 1,
and this is clearly a polynomial expression in the aij’s. (Note that wehave identified Matn(k) with kn
2here.) Moreover, the multiplication of
matrices in SLn(k) is given by n2 polynomials, as is the inverse (becausethe determinant of the matrices in SLn(k) is 1).
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(2) GLn.
If A = (aij) ∈ Matn(k) is an arbitrary matrix, then A belongs to GLn(k)if and only if
detA =∑
σ∈Symn
sgn(σ) · a1,σ(1) · · · an,σ(n) 6= 0.
This doesn’t look like a polynomial equation in the aij’s. Moreover, theinverse looks problematic because it is given by a rational function. Luck-ily, we can actually solve both problems simultaneously by Rabinowitch’strick :
GLn(k) ={
(aij, d) ∈ kn2+1 | det(aij) · d = 1}.
Observe that d plays the role of the inverse of the determinant of A; inparticular, the inverse of A now involves multiplying with d, which resultsin a polynomial expression again.
(3) Gm.
The group Gm is defined as GL1, and it is simply called the multiplicativegroup of k. Notice that the formula from above now simplifies to theform
Gm(k) ={
(s, t) ∈ k2 | st = 1}.
(4) Ga.
The group Ga is called the additive group of k, and is defined by
Ga(k) = k
(as a variety) with the addition in k as group operation. It is obvious thatthis is an affine algebraic group. In order to view it as a linear algebraicgroup, the addition has to correspond to matrix multiplication, whichcan be realized by the isomorphism
(k,+) ∼= {( 1 a1 ) | a ∈ k} .
1.2 The building bricks
We will now give an overview of five different types of algebraic groups, fromwhich all other algebraic groups are built up. For the sake of simplicity, wewill assume that char(k) = 0.
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1.2.1 Finite algebraic groups
Every finite group G can be realized as a subgroup of some GLn(k), via
GCayley rep.↪−−−−−−→ Symn
permutation mat.↪−−−−−−−−−→ GLn(k).
The group G is indeed defined by polynomials, simply because it is a finiteset. Indeed, a single element g ∈ G can clearly be defined by n2 linear equa-tions, and a finite union of something that can be described with polynomialequations, can again be described with polynomial equations1.
Such finite algebraic groups will be called constant finite algebraic groups.
1.2.2 Abelian varieties
Whereas affine algebraic groups are those algebraic groups that can be em-bedded into affine space, abelian varieties are algebraic groups that can beembedded into projective space.
Definition 1.2.1. An algebraic group is connected if it does not admit propernormal subgroups of finite index, or equivalently, if every finite quotient istrivial.
Definition 1.2.2. An abelian variety is a connected algebraic group whichis projective as an algebraic variety.
The one-dimensional abelian varieties are precisely the elliptic curves.Abelian varieties are related to the integrals studies by Abel, and it is ahappy coincidence that all abelian varieties are commutative2.
1.2.3 Semisimple linear algebraic groups
Definition 1.2.3. Let G be a connected linear algebraic group. Then G issimple if G is non-abelian and does not admit any proper non-trivial algebraicnormal subgroups. The groupG is called almost simple or quasisimple if Z(G)is finite and G/Z(G) is simple.
Example 1.2.4. The group SLn (with n > 1) is almost simple. Indeed, thecenter
Z ={( a ...
a
) ∣∣∣ an = 1}
is finite, and PSLn = SLn/Z is simple.1More formally, a finite union of algebraic varieties is again an algebraic variety; see
Chapter 4 later.2This is a non-trivial fact, depending on the fact that a projective variety is complete.
See also Definition 10.2.1 below.
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Definition 1.2.5. Let G,H be linear algebraic groups. An isogeny from G toH is a surjective morphism ϕ : G� H with finite kernel. Two linear algebraicgroups H1 and H2 are called isogenous if there exists a linear algebraic groupG and isogenies H1 � G � H2. Being isogenous is an equivalence relation(exercise!).
The following classification result will certainly look very mysterious atthis point, and one of the main goals of this course is precisely to understandthe meaning of this major theorem.
Theorem 1.2.6. Let k be an algebraically closed field with char(k) = 0. Thenevery almost simple linear algebraic group over k is isogenous to exactly oneof the following.
classical types
An
Bn
Cn
Dn
exceptional types
E6, E7, E8
F4
G2
The groups of type An are groups isogenous to the special linear group SLn+1;the groups of type Bn are groups isogenous to the orthogonal group SO2n+1;the groups of type Cn are groups isogenous to the symplectic group Sp2n; thegroups of type Dn are groups isogenous to the orthogonal group SO2n.
Definition 1.2.7. A linear algebraic group G is an almost direct product ofits subgroups G1, . . . , Gr if the product map
G1 × · · · ×Gr → G
(g1, . . . , gr) 7→ g1 · · · gr
is an isogeny.
Example 1.2.8. The groupG = (SL2×SL2)/N whereN = {(I, I), (−I,−I)}is an almost direct product of SL2 and SL2. Note, however, that it is not adirect product of almost simple subgroups.
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Definition 1.2.9. A linear algebraic group G is semisimple if it is an almostdirect product of almost simple subgroups.
We will later see a very different (but equivalent) definition in terms ofthe radical of the group; see Definition 11.1.2 below.
Remark 1.2.10. The group GLn is not semisimple, but as we will see in amoment, it is a so-called reductive group; these groups are not too far frombeing semisimple (in a precise sense).
1.2.4 Groups of multiplicative type and tori
Definition 1.2.11. Let T be an algebraic subgroup of GL(V ) for some n-di-mensional vector space V over k. Then T is of multiplicative type if it isdiagonalizable over the algebraic closure k, i.e. if there exists a basis forV (k) = k
nsuch that T is contained in
Dn :=
{A =
( ∗ 0...0 ∗
) ∣∣∣ A is invertible
}.
If in addition T is connected, then we call T an (algebraic) torus.
We also recall the corresponding definition for individual elements of agroup.
Definition 1.2.12. Let G ≤ GL(V ) be a linear algebraic group, and letg ∈ G(k). Then g is diagonalizable if there exists a basis for V (k) such thatg ∈ Dn(k), and g is called semisimple if it is diagonalizable over k.
1.2.5 Unipotent groups
Definition 1.2.13. Let G be an algebraic subgroup of GL(V ) for some n-di-mensional vector space V over k. Then G is unipotent if there exists a basisfor V (k) such that G is contained in
Un :=
{( 1 ∗ ∗0
... ∗0 0 1
)}.
Definition 1.2.14. Let G ≤ GL(V ) be a linear algebraic group, and letg ∈ G(k). Then g is unipotent if the following equivalent conditions aresatisfied:
(i) g − 1 is nilpotent, i.e. (g − 1)N = 0 for some N ;
(ii) the characteristic polynomial χg(t) for g is a power of (t− 1);
(iii) all eigenvalues of g in k are equal to 1.
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1.3 Extensions
1.3.1 Solvable groups
Definition 1.3.1. A linear algebraic group G is solvable if there is a chainof algebraic subgroups
G = G0 DG1 D · · ·DGn−1 DGn = 1
such that each factor Gi/Gi+1 is abelian.
Examples 1.3.2. (1) The group Un is solvable.
(2) The group
Tn :=
{A =
( ∗ ∗ ∗0
... ∗0 0 ∗
) ∣∣∣ A is invertible
}is solvable; notice that Tn/Un
∼= Dn.
The following important result (which we will come back to later) showsthat when k is algebraically closed, every connected solvable algebraic groupcan be realized as a group of upper triangular matrices.
Theorem 1.3.3 (Lie–Kolchin). Let k be an algebraically closed field, and letG ≤ GL(V ) be a connected linear algebraic group. Then G is solvable if andonly if there is a basis for V (k) such that G ≤ Tn.
1.3.2 Reductive groups
Definition 1.3.4. A connected linear algebraic group is called reductive ifit does not admit any non-trivial connected unipotent normal subgroups.
When char(k) = 0, any reductive group is an extension of a semisimplegroup by a torus:
1→ T → G→ G/T → 1,
where G/T is semisimple. This is no longer true for fields of positive char-acteristic, but it is still almost true (in a precise sense); this has recently ledto the study of so-called pseudo-reductive groups [CGP15, CP16].
Example 1.3.5. The group GLn is reductive:
1→ Gm → GLn → PGLn → 1.
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1.3.3 Disconnected groups
Recall that an algebraic group is disconnected if it admits an algebraic normalsubgroup of finite index > 1. We will give two different examples illustratingthat disconnected groups come up naturally in certain situations.
Examples 1.3.6. (1) The orthogonal group On is defined by
On(k) := {A ∈ GLn(k) | AtA = I}.
Since the determinant of an orthogonal matrix is always 1 or −1, thespecial orthogonal subgroup SOn defined by
SOn(k) := {A ∈ On(k) | detA = 1}
is a normal subgroup of index 2:
1→ SOn → Ondet−→ Z/2Z→ 1.
Therefore, the group On is not connected. (The group SOn is connected,but that is certainly a non-trivial fact.)
(2) A matrix is called monomial if it has exactly one non-zero entry on eachrow and each column. The group Monn defined by
Monn(k) := {A ∈ GLn | A is monomial}
is disconnected:
1→ Dn → Monn → Symn → 1.
Notice that this group arises naturally as the normalizer of Dn inside GLn.
1.4 Overview
We finish this introductary chapter by presenting an overview of how anarbitrary algebraic group can be decomposed into smaller pieces that we aremore likely to understand. We assume that char(k) = 0.
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algebraic group G
finite etalealg. group
connectedalg. group
abelianvariety
connected affinealg. group
reductivealg. group
solvablealg. group
semisimplealg. group
torusunipotentalg. group
G/G◦ G◦
G◦/H H
H/Ru(H) R(H)
H/R(H) Ru(H)R(H)/Ru(H)
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Chapter 2 Algebras
In order to build up the theory of linear algebraic groups, it will be essentialto have a good understanding of commutative k-algebras. We take the op-portunity to introduce the theory of k-algebras in general (not restricting tocommutative algebras), since these algebras will allow us to give interestingexamples of certain types of linear algebraic groups anyway.
We will often use K for a commutative ring (always assumed to be a ringwith 1) and k for a commutative field.
2.1 Definitions and examples
Definition 2.1.1. Let K be a commutative ring.
(i) An algebra over K or a K-algebra is a (not necessarily commutative)ring A with 1 which is also a K-module, such that the multiplicationin A is K-bilinear:
αx · y = x · αy = α(xy)
for all x, y ∈ A and all α ∈ K.
(ii) A morphism of K-algebras is a K-linear ring morphism.
(iii) A subalgebra of a K-algebra is a subring which is also a K-submodule.
(iv) A left ideal of a K-algebra A is a K-submodule I of A such that AI ⊆ I;a right ideal is defined similarly. A two-sided ideal (or simply an ideal)is a submodule which is simultaneously a left and right ideal.
(v) The center of a K-algebra A is the subalgebra
Z(A) := {z ∈ A | zx = xz for all x ∈ A}.
(vi) The natural map
η : K → A : α 7→ α · 1
is a ring morphism from K to Z(A), which is called the structure mor-phism of A.
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Remarks 2.1.2. (i) The structure morphism η is not necessarily injective,and hence K is not always a subalgebra of A. (For instance, Z/nZ isa Z-algebra.) On the other hand, if K = k is a field, then η is alwaysinjective, and then we can consider k as a subalgebra of A by identifyingk with k · 1 ⊆ A.
(ii) Given a not necessarily commutative ring A with 1 and a ring morphismη : K → Z(A), we can make A into a K-algebra by endowing it withthe K-module structure
α · x := η(α)x
for all α ∈ K and all x ∈ A; the map η is then precisely the structuremorphism of the K-algebra A.
(iii) It is also possible to define a K-algebra more generally as a K-moduleA endowed with a K-bilinear multiplication, without assuming A to bea ring. In particular, A might not have a unit 1, and A might not beassociative. If A has a unit 1, then it is called a unital K-algebra. Thesenot necessarily associative algebras turn up very often in the study ofthe exceptional linear algebraic groups. For instance, each algebraicgroup of type G2 can be realized as the automorphism group of a so-called octonion algebra, which is a certain 8-dimensional non-associativeunital algebra.Another important family of non-associative non-unital algebras is givenby the Lie algebras, which we will study in more detail in Chapter 7.
(iv) If K = k is a field, then any k-algebra A (in the general sense fromabove) is in particular a vector space over k, with some basis (ui)i∈I .In this case, the multiplication on A is completely determined by itsstructure constants γijr ∈ k:
ui · uj =∑r∈I
γijrur,
for all i, j ∈ I, and where for fixed i and j, the constants γijr arenon-zero for finitely many r ∈ I only.
Definition 2.1.3. If every non-zero element of a K-algebra A has an inverse,then we call A a skew field. Of course, this implies in particular that Z(A) isa field, and A is a k-algebra for k = Z(A). (Notice that k might be differentfrom K, however, and K is not necessarily a field.) If in addition dimk A isfinite, then we call A a division algebra or a division ring1.
1Some care is needed, since some authors do not make this distinction between skewfields and division rings. Most of the time, however, this should be clear form the context.
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Examples 2.1.4. Let K be a commutative ring.
(1) Every ring is a Z-algebra, and conversely.
(2) The set Tn(K) of all2 upper-triangular n by n matrices over K is aK-algebra, with structure morphism
η : K → Tn(K) : α 7→ diag(α, . . . , α).
(3) Let V be a K-module, and A = EndK(V ). Then A is a K-module definedby
(α · f)(x) := f(αx) for all x ∈ V,
for all α ∈ K and all f ∈ A. This K-module structure makes the ring Ainto a K-algebra.If V is free of rank n, then A ∼= Matn(K).
(4) Let M be a monoid, i.e. a set endowed with a binary associative operationwith a neutral element3. Let A be the free K-module over M , and endowA with the multiplication induced by M . Then A is a K-algebra, whichwe denote by A = KM , and which we call the monoid K-algebra inducedby M . If M = G is a group, then we call A = KG the group K-algebrainduced by G. We give some concrete examples.
(a) Let M = {1, x, x2, . . . }. Then M is a monoid which is not a group;the corresponding monoid algebra KM is isomorphic to the polyno-mial algebra K[x].
(b) Let M = 〈x〉 be an infinite cyclic group. Then the group algebraKM is isomorphic to the algebra of Laurent polynomials K[x, x−1].
(c) LetM = 〈x〉 be a cyclic group of order n. ThenKM ∼= K[x]/(xn−1).
(d) Let M be the free monoid on {x1, . . . , xn}. Then KM is called thefree associative algebra on x1, . . . , xn, and is denoted byK〈x1, . . . , xn〉.
It is an easy but important fact that every finite-dimensional k-algebracan be embedded into a matrix algebra. (This fact can be compared tothe Cayley representation for finite groups, which embeds an arbitrary finitegroup into a symmetric group.)
Definition 2.1.5. Let A be a finite-dimensional k-algebra. A (matrix) rep-resentation for A is a k-algebra morphism ρ : A→ Matr(k) for some naturalnumber r. If ρ is injective, then the representation is called faithful.
2including the non-invertible ones, so this is not the same as Tn(K) defined above.3Informally, a monoid is a group without inverses.
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Theorem 2.1.6. Let A be a finite-dimensional k-algebra. Then A is iso-morphic to a subalgebra of Matn(k), i.e. A has a faithful representation.
Proof. For each a ∈ A, the map
λa : A→ A : x 7→ ax
is an element of Endk(A) ∼= Matn(k). The corresponding map
λ : A→ Matn(k) : a 7→ λa
is an algebra morphism. Clearly, λa is the zero map only for a = 0, hencethe morphism λ is injective. �
The representation λ that we have constructed in the previous proof iscalled the left regular representation for A. Of course, one can similarly definethe right regular representation for A.
2.2 Tensor products
In our future study of algebraic varieties, the tensor product of (commuta-tive) k-algebras will be invaluable. In fact, in the category of commutative4
k-algebras, the tensor product turns out to be precisely the so-called co-product, which already illustrates its importance. But first, we will have acloser look at tensor products of K-modules in general (where K is still acommutative ring with 1).
2.2.1 Tensor products of K-modules
Tensor products are intimately related to bilinear forms, and in fact, thetensor product is, in a precise sense that we will describe below, the mostuniversal object to which a bilinear form from the pair U, V can map, in thesense that every other bilinear map factors through the tensor product.
Definition 2.2.1. Let U and V be two K-modules. The tensor product ofU and V is defined to be a pair (T, p) consisting of a K-module T and aK-bilinear map p : U × V → T , such that for every K-module W and everyK-bilinear map f : U × V → W , there is a unique K-module morphism
4It is not true that the tensor product is the coproduct in the category of all k-algebras.
12
f ′ : T → W such that f = f ′ ◦ p.
U × V T
W
p
ff ′
Notice that it is not immediately obvious that the tensor product existsat all, but as we will see in a minute, it is not too hard to see that if it exists,it is necessarily unique; we will denote T by U ⊗K V or by U ⊗ V if the ringK is clear from the context (which is not always the case!). We will rarelyexplicitly write down p, i.e. we will simply say that T = U⊗K V is the tensorproduct of U and V .
Lemma 2.2.2. Let U and V be two K-modules. If the tensor product of Uand V exists, then it is unique.
Proof. The proof will only use the universality of the defining property; thefact that U and V are K-modules will turn out to be irrelevant.
So assume that (T1, p1) and (T2, p2) are two tensor products of U and V .We first invoke the universal property for T1 to obtain a (unique) morphismf1 : T1 → T2 such that p2 = f1 ◦ p1; similarly, there is a unique morphismf2 : T2 → T1 such that p1 = f2 ◦ p2. Hence
p1 = (f2 ◦ f1) ◦ p1 and p2 = (f1 ◦ f2) ◦ p2.
We now use the universal property for T1 again, but this time with W = T1
and f = p1. By the uniqueness aspect of the universal property, we getthat f2 ◦ f1 = idT1 , and similarly f1 ◦ f2 = idT2 . We conclude that f1 isan isomorphism from T1 to T2, and hence the pairs (T1, p1) and (T2, p2) areisomorphic.
U × V
T1
T2
p1
p2
f1 f2 U × V
T1
T1
p1
p1
id f2 ◦ f1
�
We will now show how to construct the tensor product of two K-modules;this will at the same time prove the existence of the tensor product. The ideais that we first consider a free object, from which we construct the tensorproduct by modding out the required relations.
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Construction 2.2.3. Let U and V be two K-modules. Define A to be thefree K-module over the set U × V . Now consider the submodule
B :=
⟨(u+ u′, v)− (u, v)− (u′, v),
(u, v + v′)− (u, v)− (u, v′),
(αu, v)− α(u, v), (u, αv)− α(u, v)
∣∣∣∣∣ u, u′ ∈ U, v, v′ ∈ V, α ∈ K⟩.
Finally, let T := A/B, and let p : U × V → T be the composition
p : U × V ↪→ A� T.
It is not too hard to check that the pair (T, p) satisfies the universal propertydefining the tensor product, and hence T is indeed the tensor product U⊗KV .
Remarks 2.2.4. (i) We will usually write T = U ⊗K V , even though thetensor product is in principle only defined up to isomorphism. Typically,we have the above construction in mind when we write such an equality(rather than an isomorphism). In particular, the image p(u, v) of a pair(u, v) ∈ U × V under p will be written as u⊗ v.
(ii) It is a common beginners’ mistake to write an arbitrary element ofT = U ⊗K V as u⊗ v. These elements only generate T as a K-module,and hence an arbitrary element of T is a finite sum
x =∑i
ui ⊗ vi,
where ui ∈ U and vi ∈ V .
(iii) The universal property of tensor products can conveniently be rephrasedby the isomorphism
HomK(U ⊗K V,W ) ∼= HomK(U,HomK(V,W )).
This property is known as adjoint associativity.
We now list a few properties of the tensor product, the proof of which weleave to the reader.
Properties 2.2.5. Let U, V,W be K-modules. Then
(i) U ⊗ V ∼= V ⊗ U ;
(ii) U ⊗ (V ⊗W ) ∼= (U ⊗ V )⊗W ;
(iii) U ⊗ (V ⊕W ) ∼= (U ⊗ V )⊕ (U ⊗W );
14
(iv) U ⊗Kn ∼= Un;
(v) Kn ⊗Km ∼= Knm.
In fact, we can make properties (iv) and (v) more precise, as follows.
Proposition 2.2.6. Let U be a K-module and V a free K-module of rank n,with basis (e1, . . . , en). Then every element x of U ⊗K V can be uniquelywritten as
x =n∑i=1
ui ⊗ ei
with ui ∈ U .
Proof. First prove the statement for n = 1 using the universal property oftensor products. Then deduce the general statement by induction on n, usingthe distributive property 2.2.5(iii). We leave the details as an exercise. �
Definition 2.2.7. If f : U → V and g : U ′ → V ′ are two K-module mor-phisms, then we define
f ⊗ g : U ⊗K U ′ → V ⊗K V ′ : u⊗ v 7→ f(u)⊗ g(v).
By the universal property defining tensor products, this is a well definedK-module morphism.
Remark 2.2.8. If f : U → V is an injective K-module morphism, then theinduced morphism
f ⊗ id : U ⊗W → V ⊗W
is not always injective! (Consider for instance the map f : 2Z→ Z given byinclusion, and let W = Z/2Z.) If K is a field, however, then f ⊗ id remainsinjective.
In fact, this leads to an important notion: a K-module W is called flatprecisely when, for each injective morphism f : U → V of K-modules, theinduced morphism f ⊗ id : U ⊗W → V ⊗W is injective.
2.2.2 Tensor products of K-algebras
Recall that a K-algebra is a K-module equipped with a K-bilinear multipli-cation (which is not necessarily associative and does not necessarily have aneutral element). In fact, by the universal property of tensor products, wecan view the multiplication as a morphism
m : A⊗ A→ A.
15
It is interesting to express the unit element and the associativity in termsof this morphism m. The algebra A is associative if and only if the mapsm ◦ (m⊗ id) and m ◦ (id⊗m) from A⊗A⊗A to A coincide, i.e. if and onlyif the following diagram commutes:
A⊗ A⊗ A A⊗ A
A⊗ A A
m⊗ id
id⊗m m
m
On the other hand, the algebra A is unital, with unit e ∈ A, if and only if
m(e⊗ x) = x = m(x⊗ e)
for all x ∈ A. This can be expressed in a more fancy fashion, using thestructure morphism η (whose existence is equivalent to the existence of theunit e):
m ◦ (η ⊗ id) = πA = m ◦ (id⊗ η),
where πA is the natural isomorphism from K ⊗K A to A. Equivalently, thefollowing diagrams commute:
K ⊗ A
A
A⊗ A
A
η ⊗ id
πA m
A⊗K
A
A⊗ A
A
id⊗ η
πA m
With this in mind, we can give an intrinsic description of the tensorproduct of two K-algebras.
Definition 2.2.9. Let A and B be two K-algebras, with multiplication mor-phisms m and n, respectively. Let C = A ⊗K B as a K-module. In orderto make C into a K-algebra, it only remains to describe the multiplicationmorphism z. First, define5 a morphism
τ : A⊗B → B ⊗ A : a⊗ b 7→ b⊗ a5Recall that an arbitrary element of A ⊗ B is of the form
∑i ai ⊗ bi, but in order to
describe a morphism from A⊗B to a third module M , it suffices to prescribe the morphismon the set of generators {a⊗ b | a ∈ A, b ∈ B}.
16
for all a ∈ A and all b ∈ B. Now let
z := (m⊗ n) ◦ (idA ⊗ τ ⊗ idB) : A⊗B ⊗ A⊗B → A⊗B.
Explicitly, if a1, a2 ∈ A and b1, b2 ∈ B, then the multiplication satisfies
(a1 ⊗ b1) · (a2 ⊗ b2) = a1a2 ⊗ b1b2,
and by K-bilinearity, this formula also uniquely defines the multiplication oftwo arbitrary elements of A⊗B.
Examples 2.2.10. (1) Let A be an arbitrary K-algebra. We claim that
A⊗K Matn(K) ∼= Matn(A).
Indeed, note that this isomorphism certainly holds as K-modules. ByProposition 2.2.6, every element of A⊗Matn(K) can be uniquely writtenas
x =n∑
i,j=1
aij ⊗ eij
with aij ∈ A, and where (eij) is the canonical basis of Matn(K). By thedefinition of the tensor product of K-algebras, it is now clear that theK-module isomorphism
ϕ : A⊗Matn(K)→ Matn(A) :n∑
i,j=1
aij ⊗ eij 7→ (aij)
is indeed a K-algebra isomorphism.
(2) A special case of the previous example is obtained if A is itself a fullmatrix algebra:
Matr(K)⊗K Matn(K) ∼= Matrn(K).
(3) Consider the polynomial algebra K[x] in one variable. Then
K[x]⊗K K[x] ∼= K[x, y],
the polynomial algebra over K in two variables.
(4) Another important example is given by extension of scalars. Let A bea k-algebra (where k is a field), and suppose that E/k is a field exten-sion. Then A⊗k E is not only a k-algebra, but also an E-algebra, withdimE(A ⊗k E) = dimk A. This algebra is often simply denoted by AE.Notice that by Proposition 2.2.6, if (e1, . . . , en) is a basis for A as ak-vector space, then (e1⊗ 1, . . . , en⊗ 1) is a basis for AE as an E-vectorspace.
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Chapter 3 Categories
Category theory often looks quite daunting when first encountered. It is atheory that looks too abstract to be meaningful. However, as we will see,it is actually very powerful when used appropriately. It is not only a useful“language”, but it also allows to switch from one interpretation to anotherin a mathematically rigorous fashion.
We will later introduce linear algebraic groups as functors, which go fromone category to another. This will allow us to switch viewpoints betweenlinear algebraic groups as group functors on the one hand, and Hopf algebrason the other hand. The abstract tool that will connect these two viewpointsis the Yoneda Lemma, which is sometimes referred to as a “deep triviality”.
3.1 Definition and examples
Definition 3.1.1. A category C consists of a class ob(C) of objects and aclass mor(C) or hom(C) of morphisms. Each morphism α ∈ hom(C) has twoassociated objects, called the source (X ∈ ob(C)) and the target (Y ∈ ob(C)),and we write
α : X → Y or Xα−−→ Y.
The class of all morphisms with source X and target Y will be denoted byhom(X, Y ). Moreover, the category C comes equipped with a composition ofmorphisms
hom(X, Y )× hom(Y, Z)→ hom(X,Z) : (α, β) 7→ α · β = αβ.
(We sometimes write β ◦ α for αβ.) In order to be a category, the objects,morphisms and composition have to satisfy the following two axioms:
Associativity of composition. If Xα−−→ Y
β−−→ Zγ−−→ T , then (αβ)γ =
α(βγ).
Identity morphisms. For each X ∈ ob(C) there is an idX ∈ hom(X,X)such that for each X
α−−→ Y we have idX · α = α = α · idY .
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Remarks 3.1.2. (i) Observe that ob(C) and hom(C) are classes and notsets. This is important from a set-theoretic point of view, but we willnot have to worry about these subtleties. It is worth pointing out thatmany categories are locally small in the sense that hom(X, Y ) is a setfor all X, Y ∈ ob(C). If this is the case, then it is natural (and oftenpart of the definition) to require1 in addition that
hom(X, Y ) ∩ hom(X ′, Y ′) = ∅ unless X = X ′ and Y = Y ′.
(ii) It is customary to say that a morphism α ∈ hom(X, Y ) is a morphismfrom X to Y . Some care is needed, however, since morphisms mightbehave very differently from ordinary maps in the set-theoretic sense.
Examples 3.1.3. (1) We list some common categories.
Name Objects Morphisms
Set sets mapsGrp groups group morphismsAbGrp abelian groups group morphismsTop topological spaces continuous mapsTop0 top. spaces with base pt. cont. maps preserving base pts.ModR right R-modules R-module morphisms
RMod left R-modules R-module morphismsRing rings with 1 ring morphisms preserving 1Rng rings ring morphismsVeck vector spaces over k linear mapsk-alg comm. assoc. k-algebras algebra morphisms
(2) There exist categories of a very different nature. For instance, let M bean arbitrary monoid. Then we can view M as a category with one object(often denoted by ∗), such that the morphisms in the category correspondto the elements of M and composition of morphisms corresponds to themonoid operation in M .
Definition 3.1.4. (i) Let Xα−−→ Y . A morphism β : Y → X such that
αβ = idX and βα = idY is called an inverse for α. The inverse of α isunique if it exists, and is then denoted by α−1. In this case, α is calledan isomorphism, and X and Y are isomorphic objects.
(ii) A category C is called small if both ob(C) and hom(C) are sets (ratherthan classes). For instance, the categories from Example 3.1.3(2) aresmall.
1Observe that this requirement only makes sense because hom(X,Y ) and hom(X ′, Y ′)are sets, and hence can be intersected.
20
(iii) A subcategory of a category C is a collection of objects and morphismsfrom C that form a category under the composition of C. In particular,if D is a subcategory of C, then
homD(X, Y ) ⊆ homC(X, Y )
for all X, Y ∈ ob(D).
(iv) If D is a subcategory of C such that
homD(X, Y ) = homC(X, Y )
for all X, Y ∈ ob(D), then we call D a full subcategory of C. Forinstance, AbGrp is a full subcategory of Grp.
(v) If C is a category, then we can define its opposite category Cop by “re-versing the arrows”: ob(Cop) = ob(C), and for all X, Y ∈ ob(C), wedeclare
homCop(Y,X) := homC(X, Y ).
For clarity, we denote the morphism in Cop corresponding to the mor-phism α ∈ hom(X, Y ) by αop ∈ hom(Y,X). The composition in Cop isalso reversed: (αβ)op = βopαop for all suitable α, β ∈ hom(C). Observethat for such a category, the typical intuition of elements of hom(X, Y )as “morphisms from X to Y ” is meaningless!
3.2 Functors and natural transformations
Informally, functors are morphisms between categories. We distinguish be-tween “arrow preserving” (covariant) and “arrow reversing” (contravariant)functors.
Definition 3.2.1. Let C,D be two categories.
(i) A (covariant) functor F from C to D is a map2 associating with eachobject X ∈ ob(C) an object F (X) ∈ ob(D), and with each morphismα ∈ homC(X, Y ) a morphism F (α) ∈ homD(F (X), F (Y )), such that:
• F (αβ) = F (α)F (β) (whenever this makes sense);
• F (idX) = idF (X) for all X ∈ ob(C).2Formally, a functor F is a pair of maps (Fob, Fhom), where Fob : ob(C) → ob(D) and
Fhom : hom(C)→ hom(D).
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(ii) A contravariant functor F from C to D is a map associating with eachobject X ∈ ob(C) an object F (X) ∈ ob(D), and with each morphismα ∈ homC(X, Y ) a morphism F (α) ∈ homD(F (Y ), F (X)), such that:
• F (αβ) = F (β)F (α) (whenever this makes sense);
• F (idX) = idF (X) for all X ∈ ob(C).
Examples 3.2.2. (1) Let F : Ring → AbGrp be given by mapping eachring (R,+, ·) to the corresponding additive group (R,+), and each ringmorphism to the corresponding group morphism. Such a functor is calleda forgetful functor since it “forgets” some of the underlying information(in this case the multiplication).
(2) Let F : Grp → Grp be given by mapping each group G to its derivedsubgroup [G,G], and each morphism to its restriction to the derivedsubgroup. Then F is a covariant functor.
(3) There is no functor F : Grp → Grp with the property that each groupG is mapped to its center Z(G). (This is an interesting exercise; this isnot completely obvious at first sight.)
(4) Let k be a field. There is a contravariant functor F : Veck → Veck whichassigns to each vector space its dual, and to each linear transformationits dual (or transpose) transformation.
(5) Let G be a group, and let C be the corresponding category with a sin-gle object ∗, as defined in Example 3.1.3(2). Then a covariant functorF : C → Set assigns a set F (∗) to the object ∗, and assigns to eachmorphism in C (i.e., to each element g ∈ G) a map F (g) from F (∗) toitself. Since g is an invertible morphism in C, also F (g) is an invert-ible morphism in Set, in other words, F (g) is a permutation of F (∗).Since F (gh) = F (g)F (h) for all g, h ∈ G, we see that F describes apermutation representation of G.Conversely, every permutation representation of G gives rise to a functorfrom C to Set.
Remark 3.2.3. Very often, we will write down functors by indicating whatthey do on objects and assume that it is clear what they do on morphisms.To emphasize this, it is customary to use the notation . For instance, thefunctor from Example 3.2.2(2) will be denoted by
F : Grp→ Grp : G [G,G].
We now go one step further in the abstractness, and we will introducenatural transformations, which are some kind of morphisms between functors.
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Definition 3.2.4. (i) Let C,D be two categories, and F,G two covariantfunctors from C to D. A natural transformation T from F to G is afamily of D-morphisms
TX : F (X)→ G(X)
such that for each Xα−−→ Y , the diagram
F (X) F (Y )
G(X) G(Y )
F (α)
TX TY
G(α)
commutes. (The definition for natural transformations between con-travariant functors is similar.)
(ii) A natural transformation which has an inverse (in the obvious sense),is called a natural isomorphism. If T : F → G is a natural isomorphismbetween the functors F and G, then F and G are called isomorphic.
(iii) Two categories C,D are isomorphic if there exist functors F : C → Dand G : D → C such that FG = 1C and GF = 1D. Although thisdefinition looks natural, this notion is (too) restrictive.
(iv) Two categories C,D are equivalent if there exist functors F : C → Dand G : D → C such that FG and 1C are isomorphic and GF and 1Dare isomorphic.
(v) Two categories C,D are anti-equivalent or dual if there exist contravari-ant functors F : C → D and G : D → C such that FG and 1C areisomorphic and GF and 1D are isomorphic.
Example 3.2.5. The categories AbGrp and ModZ are isomorphic. Onthe other hand, consider the categories FVeck of finite-dimensional vectorspaces over k, with linear transformations as morphisms, and the categoryFColk of the column spaces kn for all finite n, with linear transformations asmorphisms. Then FVeck and FColk are certainly not isomorphic (indeed,FColk is a small category but FVeck is not), but they are equivalent. (Workout the details as an exercise. This becomes easier if you use Lemma 3.2.8below.)
Definition 3.2.6. Let F : C → D be a functor. Then for each pair of objectsX, Y ∈ ob(C), the functor F induces a map
FX→Y : homC(X, Y )→ homD(F (X), F (Y )).
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(i) The functor F is faithful if FX→Y is injective for all X, Y ∈ ob(C).(ii) The functor F is full if FX→Y is surjective for all X, Y ∈ ob(C).
(iii) The functor F is fully faithful if FX→Y is bijective for all X, Y ∈ ob(C).(iv) The functor F is dense (also called essentially surjective) if each Y ∈
ob(D) is isomorphic to an object F (X) for some X ∈ ob(C).(v) The functor F is an equivalence if and only if F is fully faithful and
dense.
Definition 3.2.7. The notions “faithful”, “full”, “fully faithful” and “dense”can be defined similarly for a contravariant functor F : C → D. The functorF is then called a duality or an anti-equivalence if F is fully faithful anddense.
Lemma 3.2.8. (i) Two categories C and D are equivalent (in the sense ofDefinition 3.2.4(iv)) if and only if there exists an equivalence from Cto D.
(ii) Two categories C and D are dual (in the sense of Definition 3.2.4(v))if and only if there exists a duality from C to D.
Proof. This is left to the reader as a rather lengthy exercise. �
Lemma 3.2.9. Let C,D be two categories, and let F be a fully faithful functorfrom C to D. Then F is injective on the isomorphism classes of objects, i.e.
F (X) ∼= F (Y ) ⇐⇒ X ∼= Y
for all X, Y ∈ ob(C).
Proof. Assume that F (X) ∼= F (Y ); then there exist morphisms
F (X)γ−−→ F (Y ) and F (Y )
δ−−→ F (X)
such that γδ = idF (X) and δγ = idF (Y ). Since F is full, γ = F (α) andδ = F (β) for certain morphisms
Xα−−→ Y and Y
β−−→ X.
It follows that αβ and idX are two morphisms from X to X which are mappedby F to the morphism idF (X); since F is faithful, it follows that αβ = idX .Similarly βα = idY , and we conclude that X ∼= Y . �
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3.3 The Yoneda Lemma
The Yoneda Lemma is a quite general abstract result in category theory thatalmost looks too abstract to be useful, but as we will see, it has very powerfulconsequences. The idea is that instead of studying the category C directly,we study the functors from C to Set, and Set is of course a category that weunderstand very well. We can think of such a functor as a “representation”of C, very much like the Cayley representation tells us how we can understanda group3 by considering it as a collection of permutations, i.e. a collection ofisomorphisms in the category Set.
This general idea of a representation is caught by the notion of repre-sentable functors, which we now define.
Definition 3.3.1. Let C be a locally small4 category.
(i) Every object A ∈ ob(C) defines a functor
hA : C → Set
given on objects by
X 7→ homC(A,X)
and on morphisms by(X
f−−→ Y)7→
(homC(A,X)→ homC(A, Y )
g 7→ g · f
).
(ii) A functor F : C → Set is called representable if there is an object A ∈ob(C) such that F is isomorphic to hA; we say that F is represented bythe object A. As we will see in Corollary 3.3.4(ii) below, a representablefunctor is represented by a unique object (up to isomorphism).
(iii) Every morphism Bα−−→ A defines a natural transformation
Tα : hA → hB
via
TαX : hA(X) = homC(A,X)→ hB(X) = homC(B,X) : g 7→ α · g.3Remember that every group can be made into a category with a single object, and
in this sense the Yoneda Lemma is really a generalization of Cayley’s Theorem. SeeExample 3.2.2(5).
4Recall that C is locally small if the classes homC(A,B) are sets for all objects A,B.This condition is necessary to ensure that the functors hA end up in Set.
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This is indeed a natural transformation:
hA(X) hA(Y )
hB(X) hB(Y )
hA(f)
TαX TαY
hB(f)
g g · f
α · g (α · g) · f = α · (g · f)
Conversely, if T : hA → hB is a natural transformation, then
α := TA(idA) ∈ hB(A)
defines a morphism Bα−−→ A.
(iv) More generally, let F : C → Set be a functor, and A ∈ ob(C). Everynatural transformation T : hA → F defines an element
aT := TA(idA) ∈ F (A).
Conversely, for each a ∈ F (A) we define a natural transformationT a : hA → F given by
T aX : hA(X)→ F (X) : g 7→ F (g)(a). (3.1)
Observe that g ∈ homC(A,X) and hence F (g) ∈ homSet(F (A), F (X)),so the element F (g)(a) does indeed belong to F (X). Check for yourselfthat T a is a natural transformation.
We are now ready to state the Yoneda Lemma.
Theorem 3.3.2 (The Yoneda Lemma). Let C be a locally small category,F : C → Set be a functor, and A ∈ ob(C). The map
ξ : Nat(hA, F )→ F (A) : T 7→ aT
is a bijection; its inverse is given by the map
ψ : F (A)→ Nat(hA, F ) : a 7→ T a.
This bijection is natural both in A and in F .
26
Proof. We first show that ξ · ψ is the identity. So let T ∈ Nat(hA, F ) bearbitrary; then for each g ∈ hA(X) = homC(A,X), we have a commutativediagram
hA(A) hA(X)
F (A) F (X)
hA(g)
TA TX
F (g)
idA g
aT F (g)(aT ) = TX(g)
(3.2)
showing that each TX coincides with T aTX , and hence the natural transforma-tions T and T aT are equal.
Next, we show that ψ · ξ is the identity. So let a ∈ F (A) be arbitrary;then
aTa = T aA(idA) = F (idA)(a) = idF (A)(a) = a,
proving that ψ · ξ = idF (A) as claimed.
We now show that ξ is natural in A, i.e. if Af−−→ B, then the following
diagram has to commute.
Nat(hA, F ) F (A)
Nat(hB, F ) F (B)
ξ
F (f)
ξ
T aT
T bT?= F (f)(aT )
where the natural transformation T is obtained from T by compositionwith f , i.e. for each object X we have
TX : hB(X)→ F (X) : g 7→ TX(fg).
Observe that F (f)(aT ) = TB(f) by the previous commutative diagram (3.2).On the other hand,
bT = TB(idB) = TB(f · idB) = TB(f),
27
proving that the diagram does indeed commute.
We finally show naturality in F , i.e. if F and G are two functors from Cto Set, and U ∈ Nat(F,G) is a natural transformation from F to G, thenthe following diagram has to commute:
Nat(hA, F ) F (A)
Nat(hA, G) G(A)
ξ
· U UA
ξ
T aT
T aT?= UA(aT )
where the natural transformation T is obtained from T by compositionwith U , i.e. for each object X we have
TX : hA(X)→ G(X) : g 7→ UX(TX(g)).
It follows that
aT = TA(idA) = UA(TA(idA)) = UA(aT ),
proving that the diagram does indeed commute, and finishing the proof ofthe theorem. �
An important special case of the Yoneda Lemma is given by the followingcorollary, which establishes some kind of duality between morphisms andnatural transformations.
Definition 3.3.3. Let C be a locally small category. The functor category C∨(also denoted by SetC) is the category with as objects the functors from C toSet, and as morphisms the natural transformations between these functors.Its full subcategory of representable functors is sometimes denoted by C∨rep.
Corollary 3.3.4. Let C be a locally small category.
(i) For each pair of objects A,B ∈ ob(C), there is a natural bijection
Nat(hA, hB) ' homC(B,A).
In particular, there is a contravariant fully faithful functor
C → C∨ : A hA,
and hence C and C∨rep are dual categories.
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(ii) The functors hA and hB are isomorphic if and only if A and B areisomorphic objects.
Proof. The first statement is nothing else than the Yoneda Lemma withF = hB. The second statement now follows from Lemma 3.2.9. �
Remark 3.3.5. A particularly colorful way to express what Yoneda’s Lemmadoes, is given by the following quote (due to Ravi Vakil) that I saw onMathOverflow:
You work at a particle accelerator. You want to understand someparticle. All you can do are throw other particles at it and seewhat happens. If you understand how your mystery particle re-sponds to all possible test particles at all possible test energies,then you know everything there is to know about your mysteryparticle.
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Chapter 4 Algebraic geometry
It is of course impossible to give a decent introduction to algebraic geometryin a single chapter of this course, but luckily the amount of algebraic geom-etry that we will require is rather limited. In particular, we will mainly bedealing with affine varieties and affine schemes, and we will have little needfor developing the general theory of varieties or schemes formed by gluingtogether affine parts through the machinery of sheaves.
On the other hand, we will need to develop the basic intuition behind al-gebraic geometry, which consists precisely of relating algebraic objects (com-mutative k-algebras) and geometric objects (affine varieties, or more gener-ally, affine schemes). This fundamental relationship will be continued andenriched when we will be dealing with affine algebraic groups in the nextchapter.
4.1 Affine varieties
During this section, we will assume that k is a commutative field, and allrings will be assumed to be commutative rings with 1. We will be dealingwith the category
k-alg := category of commutative, associative k-algebras with 1.
Let An be the k-algebraAn := k[t1, . . . , tn]
of polynomials over k in n variables; we can think of An geometrically as thealgebra of k-valued functions on the affine space kn.
Definition 4.1.1. (i) An affine variety1 is a subset of kn defined as thecommon zeroes of a collection of polynomials in An. More precisely,let S ⊆ An be any collection of polynomials; then we define the affinevariety V (S) as
V (S) := {x ∈ kn | f(x) = 0 for all f ∈ S}.1Some authors require affine varieties to be irreducible, and refer to our affine varieties
as (affine) algebraic sets instead.
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(ii) Conversely, if X ⊆ kn is an arbitrary subset of the affine space kn, thenwe set
I(X) := {f ∈ An | f(x) = 0 for all x ∈ X};
then I(X) is an ideal in An, which we refer to as the ideal of polynomialsvanishing on X.
Every affine variety is defined by a finite number of polynomials:
Theorem 4.1.2 (Hilbert’s Basis Theorem). Every ideal I E An is finitelygenerated.
Proof omitted. �
Lemma 4.1.3. Let I, J be two ideals in An, and let C be any collection ofideals in An. Then:
(i) if I ⊆ J , then V (I) ⊇ V (J);
(ii) V (I) ∪ V (J) = V (I ∩ J) = V (IJ);
(iii)⋂I∈C V (I) = V (〈I | I ∈ C〉);
(iv) V ((0)) = kn and V ((1)) = ∅.
Proof. The only not completely trivial part is (ii). First, observe that I andJ both contain I ∩ J , which in turn contains IJ ; so (i) implies that
V (I) ∪ V (J) ⊆ V (I ∩ J) ⊆ V (IJ).
Now let x ∈ V (IJ) be arbitrary, and assume that x 6∈ V (I). Then thereis some f ∈ I with f(x) 6= 0. On the other hand, for each g ∈ J we havefg ∈ IJ , and hence f(x)g(x) = 0. It follows that g(x) = 0 for each g ∈ J ,hence x ∈ V (J). �
An immediate consequence of the previous lemma is that the affine vari-eties make the affine kn into a topological space.
Definition 4.1.4. The sets of the form V (I) ⊆ kn are the closed sets of atopology on kn, which we call the Zariski topology. (Indeed, Lemma 4.1.3tells us that the union of a finite number of closed sets is again closed, thatthe intersection of an arbitrary collection of closed sets is again closed, andthat the empty space and the whole space are closed.) Moreover, every affinevariety V (I) inherits this topology of kn, which we also refer to as the Zariskitopology on V (I).
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Lemma 4.1.5. The open sets of the form
D(f) := {x ∈ kn | f(x) 6= 0}
for f ∈ An form a basis for the Zariski topology on kn; these sets D(f) arecalled the principal open sets (or simply the principal opens).
Proof. This follows from the definitions, since every closed set can be writtenas the intersection of closed sets of the form V (f), and hence every open setcan be written as the union of open sets of the form D(f). (In fact, byHilbert’s Basis Theorem 4.1.2, every open set is the finite union of principalopens.) �
Notice that for every set of polynomials S ⊆ An, we have the inclusionS ⊆ I(V (S)), and conversely, for every subset X ⊆ kn of the affine space,the inclusion X ⊆ V (I(X)) holds. It is natural to ask when equality holds.Of course, we have X = V (I(X)) precisely when X is an affine variety, i.e.when X is closed in the Zariski topology; in fact, for general X ⊆ kn, the setV (I(X)) is precisely the closure X of X in the Zariski topology.
The question when S = I(V (S)) is more interesting. Assume that I isan ideal of the form I(V ) for some subset V ⊆ kn. Observe that I has theproperty that whenever fm ∈ I for some f ∈ An and some m ≥ 1, thenf ∈ I. Such an ideal is called a radical ideal.
Definition 4.1.6. Let I ER be an ideal in some ring2 R. Define the radicalof I as the ideal
rad I :=√I := {r ∈ R | rm ∈ I for some m ≥ 1}.
The ideal I is called a radical ideal when I = rad I.
Radical ideals in noetherian rings behave nicely:
Theorem 4.1.7 (Lasker–Noether Theorem). Every radical ideal I in a noethe-rian ring is the intersection of a finite number of prime ideals. Moreover,there is a unique irredundant3 intersection into prime ideals up to reordering.
Proof omitted. �
Remark 4.1.8. The Lasker–Noether Theorem states more generally thatevery ideal in a noetherian ring R is a finite intersection of primary ideals,i.e. of ideals I such that in the ring R/I, each zero divisor is nilpotent. Wewill not need this more general statement.
2Remember that our rings are commutative rings with 1.3An intersection of sets A1 ∩ · · · ∩An is called irredundant if removing any of the Ai’s
changes the intersection.
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For algebraically closed fields, the observation that the ideals of the formI(V ) are radical is the only required restriction.
Theorem 4.1.9 (Hilbert’s Nullstellensatz). Let k be an algebraically closedfield, and let I E An be an ideal. Then I(V (I)) = I if and only if I is aradical ideal.
Proof. We will omit the proof. There exist various rather different proofs;notice that by the Lasker–Noether Theorem, it suffices to consider the casewhen I is a prime ideal. �
Remark 4.1.10. The assumption for k to be algebraically closed, is essential.For instance, consider the principal ideal I = (x2 + y2 + 1) in R[x, y]. ThenV (I) = ∅, and hence I(V (I)) = R[x, y] 6= I.
Corollary 4.1.11. Let k be an algebraically closed field. The rule
I 7→ V (I)
is an inclusion-reversing bijection between radical ideals in An and affinevarieties in kn. Maximal ideals correspond to points, and are thus of theform
M = (t1 − a1, . . . , tn − an).
Every affine variety can be decomposed into irreducible affine varieties.In order to make this notion precise, we have to introduce some topologicalterminology.
Definition 4.1.12. Let X be a topological space.
(i) We call X connected if there are no non-empty open subspaces U1, U2 ⊆X such that X = U1 ∪ U2 and U1 ∩ U2 = ∅.
(ii) We call X irreducible if there are no non-empty open subspaces U1, U2 ⊆X such that U1∩U2 = ∅, i.e., every two non-empty open subspaces of Xintersect non-trivially.
(iii) Let U ⊆ X. Then we call U connected, resp. irreducible, if it is con-nected, resp. irreducible in the subspace topology induced by X.
(iv) An irreducible component of X is a maximal irreducible subset. Noticethat irreducible components are always closed.
Clearly every irreducible space is also connected, but the converse is nottrue.
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Example 4.1.13. Consider the real line X = R with the ordinary realtopology. Then X is connected, but is certainly not irreducible; there areplenty of open subspaces intersecting trivially.
On the other hand, consider the real line Y = R equipped with theZariski topology. Then Y is irreducible. Indeed, every closed subspace ofY is a finite set, and hence any two open subspaces are cofinite and henceintersect non-trivially.
Lemma 4.1.14. Let X be a topological space. The following conditions areequivalent:
(a) X is irreducible.
(b) X cannot be written as the union of two closed proper subsets.
(c) Every non-empty open subset of X is dense.
Proof. Exercise. �
Observe that when U is itself a closed subspace of X, then U is irreducibleif and only if U cannot be written as the union of two closed subspaces of Xdifferent from U .
When k is algebraically closed, the irreducible affine varieties correspondto prime ideals.
Lemma 4.1.15. Assume that k is an algebraically closed field. Let I be aradical ideal in An. Then V (I) is irreducible if and only if I is a prime ideal.
Proof. This follows from the Lasker–Noether Theorem 4.1.7, but we will givea direct proof instead. Assume first that V (I) is irreducible, and let f, g ∈ Ansuch that fg ∈ I. Then V = V (I) ⊆ V (fg) = V (f) ∪ V (g), i.e.
V =(V ∩ V (f)
)∪(V ∩ V (g)
).
Since V is irreducible, we have either V ⊆ V (f) or V ⊆ V (g) (or both), andhence, by Corollary 4.1.11, f ∈ I or g ∈ I.
Conversely, assume that I is prime, and that V = V (I1) ∪ V (I2) forcertain radical ideals I1 and I2. By Corollary 4.1.11 again, this implies thatI ⊆ I1 and I ⊆ I2. Assume that V 6= V (I1). Then I 6= I1, so we can picksome f ∈ I1 \ I. For all g ∈ I2, we now have fg ∈ I since fg vanishes onV (I1) ∪ V (I2); because I is prime, we must have g ∈ I. This shows thatI2 ⊆ I, and hence V = V (I2), proving that V is irreducible. �
Corollary 4.1.16. Every affine variety V is a finite union of irreducibleaffine varieties; these irreducible affine varieties are uniquely determined,and are precisely the irreducible components of V .
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Proof. This follows from the Lasker–Noether Theorem 4.1.7. �
4.2 The coordinate ring of an affine variety
In this section, we will always assume that k is an algebraically closed field.By Hilbert’s Nullstellensatz (or its Corollary 4.1.11), there is a bijective cor-respondence between affine varieties (geometric objects) and radical ideals(algebraic objects).
We will take this correspondence even further. To each affine k-variety,we will associate an algebraic object, namely its coordinate ring (which willbe a k-algebra). As we will soon see, this algebra carries all information ofthe geometric object, and in fact, we will often jump back and forth betweenthe geometric objects and the algebraic objects. We will make this corre-spondence very strong and formal: these objects will form dual categories.
Definition 4.2.1. Let V be an affine variety in kn, and let I = I(V ) E Anthe corresponding ideal of polynomials vanishing on V . Then the restrictionsof the elements of An to the set V form a ring A, called the ring of regularfunctions on V or the coordinate ring or coordinate algebra of V ; it is denotedby A = k[V ] or by A = O[V ]. Explicitly,
k[V ] = O[V ] = An/I(V ),
since two elements of An restricted to V coincide if and only if their differenceis zero on V , i.e. belongs to I(V ).
Proposition 4.2.2. Let V be an affine variety in kn, and I = I(V )E An.
(i) The k-algebra A = k[V ] is finitely generated and reduced, i.e. does notcontain non-zero nilpotent elements.
(ii) There is a bijection
V → homk-alg(A, k) : x 7→ ex,
where for each x ∈ V , the evaluation morphism ex is defined by
ex : A→ k : f 7→ f(x).
Proof. (i) The k-algebra A is a quotient of An, which is finitely generated;hence A is finitely generated as well. Assume that f ∈ A is a nilpotentelement, i.e. fN = 0 for some positive integer N . Let f be an elementof An representing f ∈ An/I; then fN ∈ I. Since I is a radical ideal, itfollows that f ∈ I, and hence f = 0.
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(ii) Observe that for each x ∈ V , the evaluation morphism ex is indeed ak-algebra morphism from A to k. We will show that the map x 7→ exis a bijection.To show injectivity, let ti ∈ An be the i-th coordinate map (for each i),and let si be the image of ti in A = An/I. We will show that theelement x is uniquely determined by the morphism ex. Indeed, letx = (x1, . . . , xn) ∈ V ; then
ex(si) = si(x) = ti(x) = xi,
and hencex =
(ex(s1), . . . , ex(sn)
).
To show surjectivity, let α ∈ homk-alg(A, k) be arbitrary, and define
x :=(α(s1), . . . , α(sn)
)∈ kn.
Let α ∈ homk-alg(An, k) be given by the composition
Anproj−−→ An/I = A
α−−→ k;
then α(si) = α(ti) for all i. For each f ∈ I, we have α(f) = 0, andhence
f(x) = f(α(t1), . . . , α(tn)
)= α(f) = 0
because α is a k-algebra morphism. We conclude that x ∈ V (I) = V ,and ex = α. �
Conversely, every finitely generated reduced k-algebra arises from anaffine variety:
Proposition 4.2.3. Let A be a finitely generated reduced k-algebra. Thenthere is an affine variety X with coordinate ring A.
Proof. Let Y be the set Y := homk-alg(A, k); we will endow Y with thestructure of an affine variety. Assume that the k-algebra A is generated bysome finite set {s1, . . . , sn}, and define
ϕ : Y → kn : α 7→(α(s1), . . . , α(sn)
).
Observe that ϕ is injective because s1, . . . , sn generate A. On the other hand,let An = k[t1, . . . , tn] and let ϕ∗ be the k-algebra morphism defined by
ϕ∗ : An → A : ti 7→ si
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for each i. Then ϕ∗ is surjective because A is generated by s1, . . . , sn. If wedenote its kernel by I, then A ∼= An/I; note that I is a radical ideal becauseA is reduced.
Let X = imϕ ⊆ kn; it remains to show that X = V (I). Notice that thiswill then also imply that I(X) = I(V (I)) = I since I is radical, and hencek[X] = An/I(X) = An/I ∼= A.
For each x ∈ kn, there is a corresponding evaluation morphism ex ∈homk-alg(An, k). Then x ∈ X = imϕ if and only if there is some α ∈ Y suchthat ex = α ◦ ϕ∗. This happens precisely when ex vanishes on the kernelof ϕ∗, i.e., when ex(I) = 0, or equivalently, when x ∈ V (I). �
The previous discussion reveals a beautiful duality between finitely gen-erated reduced k-algebras on the one hand, and affine varieties on the otherhand. Notice that we have not yet defined the notion of a morphism betweenaffine varieties. It is possible to do this directly in terms of the explicit co-ordinatization of the affine varieties, but it is nicer to use duality to get anintrinsic definition. In fact, it is also convenient to have a coordinate-freedefinition of affine varieties at hand.
Definition 4.2.4. (i) An (abstract) affine k-variety is a pair (X,A), whereX is a set, and A is a ring of k-valued functions on X, such that A isa finitely generated k-algebra, and such that the map
X → homk-alg(A, k) : x 7→ ex
(where ex is the evaluation morphism at x, as defined previously) is abijection. We often denote an abstract affine k-variety (X,A) simply byX, and we refer to A as its ring of regular functions, or its coordinatealgebra, and denote it as A = k[X] or A = O[X] as before. Note thatPropositions 4.2.2 and 4.2.3 show precisely that every affine k-varietyis also an abstract affine k-variety, and conversely. The advantage ofabstract affine k-varieties is that the definition does not refer to anembedding in some kn. Observe that the algebra A is automaticallyreduced since it is an algebra of k-valued functions.
(ii) In particular, if A is a finitely generated reduced k-algebra, we can con-sider the set X = homk-alg(A, k) —or more precisely, the pair (X,A)—as the abstract affine variety corresponding to A. Explicitly, if f ∈ A,then f is a k-valued function on X, defined by the funny looking equal-ity
f(x) := x(f) for all x ∈ X.We could think of this as a “pairing” between X and A without favoringone of the two objects as acting on the other.
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(iii) Let X and Y be two (abstract) affine k-varieties. A morphism from Xto Y is a (set-theoretic) map f from X to Y such that the correspondingdual map
f ∗ : homSet(Y, k)→ homSet(X, k) : α 7→ α ◦ f
induces a morphism from k[Y ] to k[X], which we also denote by f ∗.
Proposition 4.2.5. The abstract affine k-varieties, with morphisms as de-fined above, form a category, which is dual to the category of finitely generatedreduced k-algebras.
Proof. Let C be the category of abstract affine varieties over k and D be thecategory of finitely generated reduced k-algebras. Let F be the contravari-ant functor from C to D, mapping each affine variety X to its coordinatealgebra k[X], and each morphism f : X → Y to the corresponding morphismf ∗ : k[Y ] → k[X]. On the other hand, let G be the contravariant functorfrom D to C mapping each finitely generated reduced k-algebra A to itscorresponding abstract affine variety (X,A) with X = homk-alg(A, k) as inDefinition 4.2.4(ii), and each morphism of such k-algebras g : A→ B to thecorresponding map
g∗ : Y = homk-alg(B, k)→ X = homk-alg(A, k) : y 7→ y ◦ g.
Notice that the corresponding dual morphism g∗∗ is given by
g∗∗ : homSet(X, k)→ homSet(Y, k) : f 7→ f ◦ g∗;
we claim that this map induces a morphism from k[X] to k[Y ], which willprove that g∗ is a morphism of abstract affine varieties. We will show, infact, that g∗∗ = g on A = k[X]. Indeed, for each f ∈ A and each y ∈ Y , wehave
g∗∗(f)(y) = f(g∗(y)) = g∗(y)(f) = y(g(f)),
and since g(f) ∈ B, it follows that
g∗∗(f)(y) = g(f)(y)
for all y ∈ Y and hence g∗∗(f) = g(f) for all f ∈ A since an element of Bis uniquely determined by its values on Y . We conclude that g∗∗ = g asclaimed.
It is now straightforward to check that the functors F and G define aduality between C and D. (Use Lemma 3.2.8.) �
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To finish this section, we introduce the important notion of dimension ofa variety, and we mention some facts without proofs.
Definition 4.2.6. Let k be an algebraically closed field and let V be anaffine k-variety.
(i) When V is irreducible, its coordinate algebra k[V ] is a domain, so wecan consider its fraction field k(V ) := Frac(k[V ]). The dimension of V ,dim(V ), is defined to be the transcendence degree of k(V ) over k (i.e.,the largest possible size of an algebraically independent subset of k(V )over k).
(ii) In general, write V as a finite union V =⋃Vi of its irreducible compo-
nents. Then we define dim(V ) = max dim(Vi).
Theorem 4.2.7. Let V be an irreducible subvariety of An and let f ∈ An =k[t1, . . . , tn]. Let W = V ∩ V (f). Then either W = V , or W = ∅, or W is ahypersurface of V , which means that every irreducible component of W hasdimension dimV − 1.
Theorem 4.2.8 (Topological characterization of dimension). Suppose V isirreducible and that
V ⊃ V1 ⊃ · · · ⊃ Vd 6= ∅is a maximal chain of distinct closed irreducible subsets of V . (Maximalmeans that the chain cannot be refined.) Then dim(V ) = d.
4.3 Affine varieties as functors
Let k be an arbitrary field; we will drop our earlier restriction on k to bealgebraically closed. As we have observed, in this case the geometry doesnot carry enough information, due to the failure of Hilbert’s Nullstellensatz(think for example about the imaginary circle x2 +y2 +1 = 0 over R). Simplyextending our base field to its algebraic closure is not the right solution,since we would then lose the specific nature of our objects over the originalbase field. We would like to understand our objects over all field extensionssimultaneously, and that is where functors come into play. In fact, we will atonce allow extensions over all k-algebras, not just fields; it will soon becomeclear why we do this.
Definition 4.3.1. Let k be an arbitrary field, let I EAn = k[t1, . . . , tn], andlet A = An/I. For any R ∈ k-alg, we let
VR(I) := {x ∈ Rn | f(x) = 0 for all f ∈ I},
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and we call this the set of R-points of A. Observe that we can identifythe set VR(I) with homk-alg(A,R), in exactly the same fashion as we did inProposition 4.2.2(ii).
We are now ready to introduce the notion of affine k-functors.
Definition 4.3.2. (i) A k-functor F is a functor from the category k-algto the category Set.
(ii) Recall from Definition 3.3.1(i) that for each A ∈ k-alg, there is a cor-responding functor
hA : k-alg→ Set : R homk-alg(A,R).
A k-functor F is an affine k-functor if there exists a finitely generatedk-algebra A such that F ∼= hA; recall that A is unique up to isomor-phism by the Yoneda Lemma (see Corollary 3.3.4). We also say that Fis represented by A, and we call A the coordinate ring or the coordinatealgebra of the affine k-functor F .
Example 4.3.3. (i) Consider the k-functor
An : k-alg→ Set : R Rn.
Then An is an affine k-functor represented by An = k[t1, . . . , tn] since
Rn ∼= homk-alg(An, R)
for all R ∈ k-alg.
(ii) Let I E An, and consider the k-functor
V : k-alg→ Set : R VR(I).
Then V is an affine k-functor represented by A = An/I, precisely be-cause of the observation we made in Definition 4.3.1. The functor V issometimes called the functor of points corresponding to I.
Definition 4.3.4. Let G be an affine k-functor with coordinate algebra A,and let K/k be a field extension. Then we obtain a K-functor GK (alsodenoted by G×kK) simply by restricting the functor G to K-algebras (sinceevery K-algebra is of course also a k-algebra). Notice that
homk-alg(A,R) ' homK-alg(AK , R)
for all R ∈ K-alg, where AK = k[G] ⊗k K (see Example 2.2.10(4)). Thisimplies that GK is again an affine functor, with coordinate algebra
K[GK ] = k[G]⊗k K = AK .
This procedure is called base change or extension of scalars.
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Remark 4.3.5. Note that the coordinate algebras are no longer assumedto be reduced, i.e. they may have non-zero nilpotents. This might soundawkward from a classical point of view, but it is actually very convenient.For instance, it might very well happen that a k-algebra A is reduced, butbecomes non-reduced after base change (e.g. if A is a purely inseparable fieldextension of k, then A⊗k k will have non-zero nilpotents).
By Yoneda’s Lemma (Theorem 3.3.2), or more precisely its Corollary 3.3.4,the map A hA is a fully faithful contravariant functor from k-alg tok-func, the category of k-functors. In other words, the category of affinek-functors is anti-equivalent (i.e. dual) to the category of finitely generatedk-algebras. In particular, we do not lose any information by replacing ak-algebra A by its associated k-functor hA.
We should think of the k-functors as geometric objects: just as the affinek-varieties form a category which is dual to the category of finitely generatedreduced k-algebras when k is algebraically closed, so do the affine k-functorsform a category which is dual to the category of finitely generated k-algebras(not necessarily reduced!) when k is arbitrary. The fact that the affinek-functors hA contain enough information to recover A solves our earlierissue that the set of k-points Vk(I) alone is not rich enough.
In addition, we have gained more: we do not only have affine k-functorsat our disposal, but the whole category of k-functors. This brings us into therealm of affine schemes, even though we will not formally develop the theoryof schemes here.
We end this section with the construction of products.
Definition 4.3.6. Let F and G be two k-functors. Then the product of Fand G is the functor
F ×G : k-alg→ Set : R F (R)×G(R).
The product of two affine k-functors is again affine:
Proposition 4.3.7. Let F and G be two affine k-functors, represented by thek-algebras A and B, respectively. Then F × G is again an affine k-functor,with coordinate algebra A⊗k B.
Proof. For every k-algebra R, it follows from the universal property of tensorproducts that
F (R)×G(R) ∼= homk-alg(A,R)× homk-alg(B,R)∼= homk-alg(A⊗k B,R). �
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Chapter 5 Linear algebraic groups
We are now ready to introduce the main objects of this course. We willcontinue to adopt the functorial approach that we have started in the lastsection 4.3, and introduce affine algebraic groups as certain functors. We willsee in section 5.4 that every affine algebraic group is linear, and in fact, it ismuch more common to refer to our objects as linear algebraic groups instead.
5.1 Affine algebraic groups
Definition 5.1.1. (i) A k-group functor G is a functor G from the cat-egory k-alg to the category Grp. Every k-group functor G has anassociated k-functor GSet obtained by the composition
k-algG−−→ Grp
forget−−−→ Set.
(ii) An affine algebraic group is a k-group functor G such that the corre-sponding k-functor GSet is affine.
(iii) If G is an affine algebraic group, then GSet is represented by a uniquefinitely generated k-algebra A, which we call the coordinate ring orcoordinate algebra of G, and which we denote by k[G] or by O[G].
Our main goal in this section is to understand the additional structure onk[G] which is imposed by the fact that the k-functor arises from a k-groupfunctor.
Before we proceed, we will give some examples. Recall that, in order todescribe the functors, we will usually only describe what happens with the ob-jects and omit the description of the corresponding map between morphisms(see Remark 3.2.3).
Examples 5.1.2. (1) Define Ga as the functor
k-alg→ Grp : R (R,+).
Then for each R ∈ k-alg, we can identify Ga(R) with homk-alg(k[t], R),and hence
k[Ga] ∼= k[t],
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the ring of polynomials over k in one variable. The affine algebraic groupGa is called the additive (algebraic) group over k.
(2) Let n be a positive integer, and define SLn as the functor
k-alg→ Grp : R SLn(R).
Then SLn is an affine algebraic group with
k[SLn] ∼= k[t11, . . . , tnn] / (det(tij)− 1).
(3) Let n be a positive integer, and define GLn as the functor
k-alg→ Grp : R GLn(R).
Then by Rabinowitch’s trick (see page 2), GLn is an affine algebraic groupwith
k[GLn] ∼= k[t11, . . . , tnn, d] / (d · det(tij)− 1).
(4) The functor GL1 is also written as Gm, and called the multiplicative(algebraic) group over k. In this case,
Gm : k-alg→ Grp : R (R×, ·),
and
k[Gm] ∼= k[t, d]/(dt− 1) ∼= k[t, t−1],
the ring of Laurent polynomials over k.
(5) The functor SL1 maps every k-algebra R to the trivial group, and iscalled the trivial algebraic group over k. In this case,
k[1] ∼= k[t]/(t− 1) ∼= k.
(6) Let n be a positive integer, and define µn as the functor
µn : k-alg→ Grp : R {r ∈ R | rn = 1}.
Then µn is an affine algebraic group with
k[µn] ∼= k[t]/(tn − 1).
It is called the algebraic group of n-th roots of unity over k, and is alsoreferred to as a multiplicative torsion group. Note that k[µn] has nilpotentelements if char(k) = p | n.
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(7) Let p be a prime number, and let k be a field with char(k) = p. Wedefine the functor αp by
αp : k-alg→ Grp : R ({r ∈ R | rp = 0},+
).
Then αp is an affine algebraic group with
k[αp] ∼= k[t]/(tp).
Note that k[αp] is never reduced.
The coordinate algebra k[G], as a k-algebra, only describes the geometryof the k-functor G, not its group structure. We will now try to understandhow the group structure imposes additional structure on k[G].
Definition 5.1.3. Let G be a k-group functor. The multiplication, theinverse and the neutral element for each of the objects G(R) defines naturaltransformations
µ : G×G→ G,
ι : G→ G,
e : 1→ G.
By the Yoneda Lemma (see Corollary 3.3.4), we have corresponding k-algebramorphisms
∆: k[G]→ k[G]⊗k k[G],
S : k[G]→ k[G],
ε : k[G]→ k.
They are called the comultiplication, the antipode and the counit, respectively.
Before we will figure out which axioms these morphisms satisfy in general,we will try to get some feeling for these morphisms by some explicit examples.
Example 5.1.4. (1) Consider the additive algebraic group G = Ga : R 7→(R,+), and recall that k[Ga] ∼= k[t]. Explicitly, for each R ∈ k-alg, thereis a bijection
β : homk-alg(k[t], R)→ (R,+): α 7→ α(t).
Similary, for Ga ×Ga, we have
k[Ga ×Ga] ∼= k[t]⊗k k[t] ∼= k[t1, t2],
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and there is a bijection
γ : homk-alg
(k[t1, t2], R
)→ (R×R,+): α 7→
(α(t1), α(t2)
).
The natural transformation µ : Ga × Ga → Ga induces, for each R, amorphism
µR : homk-alg
(k[t1, t2], R
)→ homk-alg(k[t], R)
which should correspond, under the above bijections β and γ, to theaddition map from R × R to R. We express this in a commutativediagram:
homk-alg
(k[t1, t2], R
)homk-alg(k[t], R)
(R×R,+) (R,+)
µR
γ β
addition
α µR(α)
(α(t1), α(t2)
)α(t1) + α(t2) = µR(α)(t)
We conclude that
µR(α)(t) = α(t1) + α(t2) = α(t⊗ 1 + 1⊗ t)
for each R ∈ k-alg. By the Yoneda Lemma (Theorem 3.3.2), the comul-tiplication ∆ is given by
∆ = µk[G]⊗k[G]
(idk[G]⊗k[G]
),
and hence∆(t) = t⊗ 1 + 1⊗ t.
Notice that this completely determines the k-algebra morphism ∆. In acompletely similar fashion, we get
S(t) = −t and ε(t) = 0.
(2) We now consider the multiplicative algebraic group Gm : R 7→ (R×, ·),with k[Gm] = k[t, t−1]. In the same manner as in the previous example,we get
µR(α)(t) = α(t1)α(t2) = α((t⊗ 1)(1⊗ t)
)= α(t⊗ t)
46
for each k-algebra R, and hence
∆(t) = t⊗ t;
similarly,
S(t) = t−1 and ε(t) = 1.
(3) We finally consider the exampleG = GLn, with coordinate algebra k[G] =k[tij, d]/(d · det(tij)− 1). We leave the details of the computation as anexercise; the outcome is as follows:
∆(tij) =∑n
`=1 ti` ⊗ t`j,∆(d) = d⊗ d,
S(tij) = d · aji where aji is the cofactor of tji,
S(d) = det(tij),
ε(tij) = δij (the Kronecker delta),
ε(d) = 1.
We would now like to understand what conditions our k-morphisms ∆,S and ε satisfy, and once again the Yoneda Lemma will give us the answer.Let us first express the axioms of a group in terms of commutative diagramsinvolving the natural transformations µ, ι and e.
Lemma 5.1.5. Let G be a k-functor. Then G = HSet for some k-groupfunctor H if and only if there are natural transformations
µ : G×G→ G,
ι : G→ G,
e : 1→ G,
such that the following diagrams commute:
G×G×G G×G
G×G G
id× µ
µ× id µ
µ
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G× 1 G×G
G G
id× e
∼= µ
1×G G×G
G G
e× id
∼= µ
G G×G
1 G
(id, ι)
µ
e
G G×G
1 G
(ι, id)
µ
e
Proof. It is clear that for each R ∈ k-alg, the commutativity of each ofthe above diagrams translates into a similar commutative diagram for theset G(R). The first diagram expresses the associativity of each µR, thesecond and third diagram express the existence of a unit (namely eR(1))for each G(R), and the fourth and fifth diagram express the existence of aninverse map (namely ιR) in each G(R). �
Yoneda’s Lemma now immediately implies a similar statement for thecorresponding coordinate algebras.
Proposition 5.1.6. Let A be a finitely generated k-algebra, with multiplica-tion map m : A ⊗ A → A. Then A is the coordinate algebra of some affinealgebraic k-group G if and only if there are k-algebra morphisms
∆: A→ A⊗ A,S : A→ A,
ε : A→ k,
such that the following diagrams commute:
A A⊗ A
A⊗ A A⊗ A⊗ A
∆
∆ ∆⊗ id
id⊗∆
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A A⊗ A
A A⊗ k
∆
id⊗ ε∼=
A A⊗ A
A k ⊗ A
∆
ε⊗ id
∼=
A k
A⊗ A A
ε
∆ η
m ◦ (id⊗ S)
A k
A⊗ A A
ε
∆ η
m ◦ (S ⊗ id)
Proof. This follows immediately from Lemma 5.1.5 and Corollary 3.3.4. Onesubtle point is how to dualize the morphism (id, ι) : G→ G×G. Notice thatthis morphism can be decomposed as
Gdiag−−→ G×G id×ι−−→ G×G,
where diagR is the “diagonal map” G(R) → G(R) × G(R) : g 7→ (g, g).It only remains to show that the dual of diag : G → G × G is preciselym : A ⊗ A → A. This follows immediately from the Yoneda Lemma, sincediagA(idA) : A⊗ A→ A : a⊗ b 7→ idA(a)idA(b) = ab = m(a⊗ b). �
Definition 5.1.7. (i) A k-algebra A equipped with k-algebra morphisms∆, S and ε satisfying the requirements from Proposition 5.1.6 is calleda (commutative) Hopf algebra1. Explicitly, we require the followingaxioms to hold, where η : k → A is the structure morphism of thek-algebra A, and where m : A⊗ A→ A is the multiplication map:
(id⊗∆) ◦∆ = (∆⊗ id) ◦∆,
m ◦ (id⊗ ε) ◦∆ = id = m ◦ (ε⊗ id) ◦∆,
m ◦ (id⊗ S) ◦∆ = η ◦ ε = m ◦ (S ⊗ id) ◦∆.
(ii) Let A,B be two Hopf k-algebras. A Hopf algebra morphism from A toB is a k-algebra morphism f : A→ B compatible with the morphisms∆, S and ε, i.e., such that
∆B ◦ f = (f ⊗ f) ◦∆A,
SB ◦ f = f ◦ SA,εB ◦ f = εA.
1We will use the simple term Hopf algebra for a commutative Hopf algebra, but depend-ing on the context, other authors might have the more general notion of not necessarilycommutative Hopf algebras in mind.
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In fact, it is sufficient to require f to be compatible with ∆; it thenfollows that it is also compatible with S and with ε.
Corollary 5.1.8. The category of affine algebraic k-groups is anti-equivalentto the category of commutative finitely generated Hopf algebras over k.
Proof. This follows immediately from Proposition 5.1.6. �
Remark 5.1.9. Let G be an affine algebraic k-group and let R ∈ k-alg.
(i) In what follows, we will very often identify the group G(R) with theset homk-alg(A,R) without explicitly writing the bijection β as in Ex-ample 5.1.4(1).
(ii) Observe that under this identification, the unit element 1 ∈ G(k) cor-responds precisely to the counit ε : A → k; more generally, the unitelement 1 ∈ G(R) corresponds to the composition ηR ◦ ε : A → R,where ηR : k → R is the structure morphism of the k-algebra R.
Remark 5.1.10. Suppose that G and H are two affine algebraic k-groups,with coordinate algebras k[G] and k[H], respectively. By Proposition 4.3.7,the affine k-functor G×H has coordinate algebra k[G×H] ∼= k[G]⊗k k[H].In fact, G × H is again an affine algebraic k-group, and the isomorphismk[G×H] ∼= k[G]⊗k k[H] is an isomorphism of Hopf algebras. (We leave thedetails of the verification of this fact to the reader.)
We will now use this duality between the two categories to construct theso-called constant finite algebraic groups over k.
Definition 5.1.11. An affine algebraic k-group G is called finite if its coor-dinate algebra k[G] is finite-dimensional.
Example 5.1.12 (Constant finite algebraic groups). Let F be any finitegroup. Our goal is to construct an affine algebraic k-group G such thatG(R) ∼= F “as often as possible”. Let
A := homSet(F, k)
with its natural k-algebra structure. Observe that as an algebra, we simplyhave the structure of a direct product
A ∼=∏g∈F
k.
For each g ∈ F , let
eg : F → k :
{g 7→ 1,
h 7→ 0 (h 6= g).
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Then {eg | g ∈ F} forms a complete system of idempotents:
e2g = eg; egeh = 0 for all g 6= h;
∑eg = 1.
We now make A into a Hopf algebra. We define k-algebra morphisms ∆, Sand ε by setting
∆(eg) :=∑
a,b∈F | g=ab
ea ⊗ eb,
S(eg) := eg−1 ,
ε(eg) :=
{1 if g = 1,
0 if g 6= 1,
for all g ∈ F . It is now an easy exercise to verify that these morphisms satisfythe defining relations of a Hopf algebra. The associated affine algebraic groupFk is now defined by
Fk(R) = hA(R) = homk-alg(A,R).
If R is a k-algebra without non-trivial idempotents, then every k-algebramorphism from A to R necessarily maps exactly one element eg (g ∈ F ) to 1and all others to 0; we conclude that in this case, Fk(R) ∼= F , at least as a set.(Note, however, that if R does have non-trivial idempotents, then Fk(R) isalways larger than F .) We verify that for such R, the group structure inducedby ∆, S and ε coincides with the original group structure of F . Since themorphism ∆ is the dual of the natural transformation µ, the multiplicationµR is given by
µR : hA⊗A(R)→ hA(R) : f 7→ f ◦∆.
The identification between F and hA(R) is given by the bijection
β : F → hA(R) : g 7→ βg :
{eg 7→ 1,
eh 7→ 0 for all h 6= g.
Now assume that f ∈ hA⊗A(R) ∼= F × F is represented by (g1, g2) ∈ F × F ,i.e. f = (βg1 , βg2) ∈ hA(R)× hA(R). We have to show that µR(f) = βg1g2 ; itsuffices to verify this for each generator eh, i.e. we have to check whether
(f ◦∆)(eh) = βg1g2(eh)
for all h ∈ F . We leave this as an easy exercise.
The affine algebraic groups Fk are called constant finite algebraic groups.
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Remark 5.1.13. Observe that in general, the constant algebraic group Z/nis different from the algebraic group µn. For instance, over Q, the groups Z/3and µ3 are not isomorphic because they have different coordinate algebras (orsimply because Z/3(Q) has 3 elements whereas µ3(Q) has only 1 element).
However, when n is not a multiple of char(k) and k contains an n-th rootof unity, then Z/n ∼= µn (which is the case, for example, for k = C).
5.2 Closed subgroups
As for every algebraic structure, it will be invaluable to study substructures.In our setting, this means that we are interested in subgroups of affine al-gebraic groups that become affine algebraic groups in their own right. Thisbrings us to the notion of closed subgroups.
Definition 5.2.1. (i) Let C be a category, and let F be a functor from Cto Set. A functor G from C to Set is a subfunctor of F , if
• for every X ∈ ob(C), the set G(X) is a subset of F (X); and
• for every α ∈ homC(X, Y ), the morphism G(α) is the restrictionof F (α) to G(X).
(ii) Let C be a category, and let G be a functor from C to Grp. A functorH from C to Grp is a subgroup of F , if
• for every X ∈ ob(C), the group H(X) is a subgroup of G(X); and
• for every α ∈ homC(X, Y ), the morphism H(α) is the restrictionof G(α) to H(X).
(iii) If, moreover,
• H(X) is a normal subgroup of G(X) for all X ∈ ob(C),
then we call H a normal subgroup of G.
(iv) Let G be an affine algebraic k-group with coordinate algebra A = k[G].A subgroup H of G is closed (or algebraic), if H is representable by aquotient of A (as Hopf algebras).
Notice that a closed subgroup H of an affine algebraic group G is indeedagain an affine algebraic group, because k[H] is a quotient of the finitelygenerated k-algebra A, and hence is itself finitely generated.
Remark 5.2.2. Let G be an affine algebraic k-group with coordinate algebraA = k[G] and let H be a subgroup of G. If H is representable, then it is
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automatically representable by a quotient of A (and hence H is a closedsubgroup); this follows from Corollary 5.3.3 below.
Conversely, we would like to know which ideals I of A give rise to a closedsubgroup of G. This brings us to the notion of a Hopf ideal.
Definition 5.2.3. Let A be a Hopf algebra over k, and let I be an ideal ofthe k-algebra A. Then I is called a Hopf ideal of A, if
• ∆(I) ⊆ I ⊗ A+ A⊗ I;
• S(I) ⊆ I;
• ε(I) = 0.
The notion of a Hopf ideal plays the same role as the notion of an idealfor rings with respect to homomorphisms:
Lemma 5.2.4. Let ϕ : A → B be a homomorphism of Hopf algebras. Thenker(ϕ) is a Hopf ideal of A, and im(ϕ) is a Hopf subalgebra of B.
Conversely, every Hopf ideal I of A is the kernel of some homomorphismϕ : A → B of Hopf algebras, and the quotient A/I is again a Hopf algebra,isomorphic to im(ϕ).
Proof. We leave the proof of these facts as an exercise. �
The following correspondence between closed subgroups of G and Hopfideals of A is now immediate.
Corollary 5.2.5. Let G be an affine algebraic k-group with coordinate algebraA = k[G]. The closed subgroups of G are in natural one-to-one correspon-dence with the Hopf ideals of A.
Proof. This follows from Definition 5.2.1(iv) and Lemma 5.2.4. �
We give one more result for later use.
Proposition 5.2.6. Let G,H be two affine algebraic k-groups, with coordi-nate algebras A = k[G] and B = k[H], and let ϕ : G → H be a morphism.Then the kernel N = kerϕ is a normal closed subgroup of G with coordi-nate algebra k[N ] = A/IH , where IH is the augmentation ideal of B, i.e.IH is the kernel of the counit εH : B → k, and where IH := ϕ∗(IH)A is thecorresponding ideal in A.
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Proof. Notice that N is defined to be the k-group functor
N : k-alg→ Grp : R 7→ ker(G(R)
ϕR−−→ H(R)).
Then an element g ∈ G(R) = homk-alg(A,R) lies in N(R) if and only ifits composite with ϕ∗ : B → A factors through the counit εH : B → k (seeRemark 5.1.9(ii)). So let IH be the kernel of εH , and let IH := ϕ∗(IH)AEA bethe corresponding ideal in A. Then an element g ∈ G(R) = homk-alg(A,R)lies in N(R) if and only if it is zero on ϕ∗(IH), and hence on IH . We concludethat N(R) = homk-alg(A/IH , R) for all R ∈ k-alg. �
5.3 Homomorphisms and quotients
Recall that a homomorphism ϕ : G → H between affine algebraic groups isuniquely determined by its dual homomorphism ϕ∗ : k[H] → k[G] betweenHopf algebras. Perhaps surprisingly2, the injectivity and surjectivity of ϕversus ϕ∗ are related in a rather subtle fashion.
Definition 5.3.1. Let ϕ : G → H be a morphism between two affine alge-braic k-groups G and H, with dual morphism ϕ∗ : k[H]→ k[G].
(i) The morphism ϕ is called injective if ϕR : G(R)→ H(R) is injective foreach R ∈ k-alg.
(ii) The morphism ϕ is called a quotient map if ϕ∗ is injective.
The following proposition is essential, but its proof requires more theorythan we have covered (fibered products and faithful flatness of algebras).
Proposition 5.3.2. A morphism ϕ : G→ H is an isomorphism if and onlyif it is an injective quotient map.
Proof omitted. �
Corollary 5.3.3. A morphism ϕ : G→ H is injective if and only if the dualmorphism ϕ∗ : k[H]→ k[G] is surjective.
Proof. Assume first that ϕ∗ is surjective. Notice that for each R ∈ k-alg, themap ϕR : G(R) = homk-alg(k[G], R) → H(R) = homk-alg(k[H], R) is simplygiven by g 7→ g ◦ ϕ∗, which is indeed an injective map if ϕ∗ is surjective.
2The reason is that injectivity and surjectivity cannot be expressed in terms of mor-phisms in a category. The corresponding categorical notions are those of “monic” and“epic” morphisms, which for the category Set indeed amount to injectivity and surjectiv-ity, but which are genuinely different notions in many other categories.
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Conversely, assume that ϕ is injective and let A := imϕ∗, so that ϕ∗
decomposes asϕ∗ : k[H]� A ↪→ k[G].
Let G′ be the affine algebraic k-group corresponding to the Hopf algebra A;then ϕ decomposes correspondingly as
ϕ : G→ G′ → H.
Since ϕ is injective, the same is true for the map G → G′. It now followsfrom Proposition 5.3.2 that this map G → G′ is an isomorphism, and weconclude that ϕ∗ is indeed surjective. �
Remark 5.3.4. Notice that we did not give a name to morphisms ϕ forwhich each ϕR is surjective. As it turns out, this is not a very useful notion.In particular, if ϕ is a quotient map, then it is not necessarily true that eachϕR is surjective. (The converse is still true.) We give two typical examples.
(1) Let k = Q and let ϕ : Gm → Gm be the n-th power map, taking eachg ∈ Gm(R) to gn ∈ Gm(R). The dual morphism ϕ∗ : k[t, t−1] → k[t, t−1]maps t to tn; this map is clearly injective. However, the correspond-ing map ϕQ : Gm(Q) → Gm(Q) is not surjective. (On the other hand,ϕQ : Gm(Q)→ Gm(Q) is surjective.)
(2) Let ϕ : SLn → PGLn be the canonical projection. Then it can be checkedthat ϕ∗ is injective. (We will do this in the exercises for n = 2.) Onthe other hand, ϕR is in general of course not surjective; the image ofϕR is PSLn(R). In fact, PSLn is not an affine algebraic group — in otherwords, the functor G : k-alg→ Grp : R 7→ PSLn(R) is not representable.(Try to Google for “PSL is not an algebraic group” if you are interestedto know more.)
The observation about Q in the first example above is not a coincidence:
Proposition 5.3.5. Let G and H be affine algebraic k-groups and denotethe algebraic closure of k by k. Let ϕ : G→ H be a quotient map. Then themap ϕk : G(k)→ H(k) is surjective.
Proof omitted. �
Remark 5.3.6. The converse of this proposition is not true in general, butit is true whenever H is smooth; see section 8.5 below.
We will need the following proposition later; its proof again makes use offibered products.
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Proposition 5.3.7. Let ϕ : G → H be a quotient map and let N be thekernel of ϕ. Assume that ψ : G → H ′ is another homomorphism whosekernel contains N . Then ψ factors uniquely through H, i.e., there is a uniquehomomorphism ψ′ : H → H ′ such that ψ = ψ′ ◦ ϕ.
Proof omitted. �
Remark 5.3.8. When ϕ : G→ H is a quotient map with kernel N , then wecan assemble this information in a short exact sequence
1→ N → G→ H → 1.
In this case, we also denote H by G/N . We emphasize once again that thisdoes not imply that there are corresponding exact sequences
1→ N(R)→ G(R)→ H(R)→ 1
in general, and correspondingly, it is not true in general that (G/N)(R) ∼=G(R)/N(R).
It is a highly non-trivial fact that quotients by closed normal subgroupsalways exist:
Theorem 5.3.9. Let G be an affine algebraic k-group and let N be a closednormal subgroup of G. Then there exists a quotient map ϕ : G → H withkernel N ; in particular, H = G/N exists (and is unique up to isomorphism).
Proof omitted. �
5.4 Affine algebraic groups are linear
So far, we have mainly been defining the objects we are interested in, butin some sense, we have not yet proven any non-trivial theorems about affinealgebraic groups. In this section, we will use the theory we have built op sofar to show a crucial fact about affine algebraic groups, namely that they arealways linear in the sense that they can be embedded in a finite-dimensionalmatrix group. The right context to study such embeddings is representationtheory, so we will first define the necessary relevant notions.
Definition 5.4.1. (i) Let G be an affine algebraic group over k, and let Vbe a k-vector space. A representation of G is a natural transformation
ρ : G→ GLV ,
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where GLV is the k-group functor defined as
GLV (R) := GL(R⊗k V ).
(We do not require V to be finite-dimensional, although that case willeventually be of main interest.) Note that R ⊗k V is a free R-module,and GL(R⊗k V ) denotes the group of automorphisms of this R-module.
(ii) Let A be a Hopf algebra over k. An A-comodule is a pair (V,m), whereV is a k-vector space, and where m : V → A ⊗k V is a k-linear mapsuch that3
(idA ⊗m
)◦m = (∆⊗ idV ) ◦m,
(ε⊗ idV ) ◦m = idV ,
i.e. such that the following two diagrams commute.
V A⊗ V
A⊗ V A⊗ A⊗ V
m
m id⊗m∆⊗ id
V A⊗ V
V k ⊗ V
m
ε⊗ id
∼=
These two definitions are closely related:
Proposition 5.4.2. Let G be an affine algebraic k-group, with coordinatealgebra A = k[G].
(i) Let ρ : G→ GLV be a G-representation, and let m be the restriction ofρA(idA) ∈ GLV (A) = GL(A⊗k V ) to V . Then (V,m) is an A-comodule.
(ii) Conversely, let (V,m) be an A-comodule, and let ρ : G → GLV be thenatural representation given by
ρR(g) := (g ⊗ idV ) ◦m for all g ∈ G(R) = homk-alg(A,R). (5.1)
Then ρ is a G-representation.
Because of this equivalence, we will often say that a comodule (V,m) isa G-representation.
3The reader should compare the defining commutative diagrams with those of a G-mod-ule V .
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Proof. Let EndV be the k-functor
EndV : k-alg→ Set : R End(R⊗k V ).
By the Yoneda Lemma, we have a natural bijection
Nat(GSet,EndV ) ' EndV (A) = End(A⊗k V ).
Since an element of End(A⊗k V ) is uniquely determined by its restriction toa k-linear map m : V → A ⊗k V , this means that there is a one-to-onecorrespondence between natural transformation of k-functors ρ (not tak-ing into account that they arise from k-group functors!) and k-linear mapsm : V → A ⊗k V . Notice that the formula (5.1) is a direct consequence ofthe Yoneda Lemma: by equation (3.1), we have ρR(g) = EndV (g)(m) for allg ∈ G(R) = homk-alg(A,R).
It remains to show that ρ is a natural transformation of k-group functorsif and only if (V,m) is a comodule for A. In principle, this is a consequenceof the Yoneda Lemma by dualizing the axioms for a G-module, in the cat-egory k-vec of k-vector spaces, identifying a vector space V inside its dualhomk-vec(V, k), but we will give an explicit argument instead.
To do this, we will first show that ρ preserves the identity if and only if(ε ⊗ idV ) ◦ m = idV (which is the second defining identity for comodules).Indeed, notice that ρ preserves the identity if and only if ρk preserves theidentity, which holds if and only if ρk(ε) = idV . We now simply observe thatρk(ε) = (ε⊗ idV ) ◦m by equation (5.1).
We now show that ρ preserves the group multiplication if and only if(idA ⊗ m
)◦ m = (∆ ⊗ idV ) ◦ m (which is the first defining identity for
comodules). Indeed, let R ∈ k-alg, and let g, h ∈ G(R). Then gh is, bydefinition, given by the composition
A∆−→ A⊗ A (g,h)−−→ R,
and hence ρR(gh) acts on V as
Vm−→ A⊗ V ∆⊗idV−−−−→ A⊗ A⊗ V (g,h)⊗idV−−−−−→ R⊗ V.
On the other hand, ρR(g)ρR(h) acts on V as4
Vm−→ A⊗ V g⊗idV−−−→ R⊗ V idR⊗m−−−−→ R⊗ A⊗ V (idR,h)⊗idV−−−−−−−→ R⊗ V,
4Since we have defined our comodules as left comodules, we have to consider the dualaction on the right; in particular, the action of ρR(g)ρR(h) on V is given by first applyingρR(g), and then ρR(h). Some authors prefer to use right comodules, in which case thedual action is on the left.
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or equivalently, as
Vm−→ A⊗ V idA⊗m−−−−→ A⊗ A⊗ V (g,h)⊗idV−−−−−→ R⊗ V.
We conclude that ρR(gh) = ρR(g)ρR(h) for all R ∈ k-alg and all g, h ∈ G(R),if and only if
(idA⊗m
)◦m = (∆⊗ idV )◦m. (For the non-trivial implication,
choose R = A ⊗ A and let g, h be the natural inclusions as the first andsecond component, respectively, so that (g, h) = idA⊗A.) �
An important example of a representation is given by k[G] itself.
Definition 5.4.3. Let G be an arbitrary affine algebraic k-group, with coor-dinate algebra k[G] equipped with the comultiplication ∆. Then (k[G],∆) isa comodule for k[G], and hence it induces a representation of G on k[G]. Wecall this the regular representation of G. Note, however, that k[G] is almostnever finite-dimensional.
The regular representation is faithful, but this is not obvious at thispoint. (This will follow from Theorem 5.4.6 below.) We will try to finda finite-dimensional subrepresentation of the regular representation which isstill faithful.
Let us first formally define this notion:
Definition 5.4.4. Let G be an affine algebraic k-group, and let (V,m) be aG-representation.
(i) A subrepresentation of (V,m) is a k-subspace5 W ≤ V such thatm(W ) ⊆ k[G]⊗k W .
(ii) The G-representation (V,m) is locally finite if every finite-dimensionalsubspace W ≤ V is contained in some finite-dimensional subrepresen-tation.
The following lemma gives a crucial ingredient for Theorem 5.4.6 that wewant to prove in a moment.
Lemma 5.4.5. Let G be an affine algebraic k-group. Then every G-repre-sentation (V,m) is locally finite.
Proof. Let (V,m) be an arbitrary G-representation; it suffices to show thatevery v ∈ V is contained in some finite-dimensional subrepresentation. Con-sider a basis (ei)i∈I for the k-vector space k[G], and write
m(v) =∑
i ei ⊗ vi,5We will sometimes say that V is a G-representation, and not mention m explicitly;
more formally, the subrepresentation corresponding to W is the pair (W,m|W ).
59
where each vi ∈ V , and almost all vi are zero. On the other hand, we canwrite
∆(ei) =∑
j,` rij`(ej ⊗ e`),where each rij` ∈ k, and each of these sums is a finite sum. We now invokethe fact that m is a comodule for k[G]:
V k[G]⊗ V
k[G]⊗ V k[G]⊗ k[G]⊗ V
m
m id⊗m∆⊗ id
v∑ei ⊗ vi
∑ei ⊗ vi
∑∆(ei)⊗ vi =
∑ei ⊗m(vi)
It follows that ∑i,j,`
rij`(ej ⊗ e` ⊗ vi) =∑j
ej ⊗m(vj),
and comparing the coefficients of ej yields∑i,`
rij`(e` ⊗ vi) = m(vj)
for each j. We conclude that
W := 〈v, vi | i ∈ I〉
is a finite-dimensional subrepresentation of (V,m) containing v. �
We now come to the main theorem of this section.
Theorem 5.4.6. Let G be an affine algebraic group over k. Then thereexists a finite-dimensional vector space V over k and an injective morphismρ : G ↪→ GLV , so in particular G is a closed subgroup of GLV .
Proof. Recall that A = k[G] is a finitely generated k-algebra; let W be afinite-dimensional k-vector space of A that generates A (as a k-algebra). ByLemma 5.4.5, W is contained in some finite-dimensional subrepresentationV of the regular representation (A,∆); of course V still generates A as ak-algebra. Denote the corresponding natural transformation by
ρ : G→ GLV ;
60
it remains to show that ρ is injective, or equivalently, by Corollary 5.3.3, that
ρ∗ : k[GLV ]→ A
is surjective. Let {v1, . . . , vn} be a basis for V , and let
∆(vj) =n∑i=1
fij ⊗ vi,
with fij ∈ A. Then by equation (5.1), the natural transformation ρ is givenexplicitly by
ρR(g)(vj) = (g ⊗ idV )(∆(vj)) =n∑i=1
g(fij)⊗ vi
for all g ∈ G(R) ' homk-alg(A,R) and all j ∈ {1, . . . , n}, and hence ρR(g) isrepresented by the matrix
ρR(g) =(g(fij)
)ij
=(fij(g)
)ij∈ GLV (R).
It follows thatρ∗(tij) = ρA(idA)(tij) = fij
for all i, j, where tij is the (i, j)-th coordinate function, with k[GLV ] ∼=k[t11, . . . , tnn, d]/(d · det(tij)− 1). On the other hand, it follows from Defini-tion 5.1.7 that
vj = m(id⊗ ε)∆(vj) = m
(n∑i=1
fij ⊗ ε(vi)
)=
n∑i=1
ε(vi) · fij,
and hence vj ∈ im ρ∗, for all j. Since A is generated by the elements v1, . . . , vnas a k-algebra, we conclude that ρ∗ is surjective. �
We have shown that every affine algebraic group is a linear algebraicgroup, and hence we will use the common terminology “linear algebraicgroup” from now on.
Example 5.4.7. Consider the algebraic group Ga over k, with coordinatealgebra A = k[t], and choose W = 〈t〉 as a generating k-subspace of A. ThenW is contained in the subrepresentation V = 〈1, t〉 of (A,∆), and
∆(1) := 1⊗ 1,
∆(t) := t⊗ 1 + 1⊗ t.
61
We can now immediately read off the corresponding matrix, which is
(fij)
=
(1 t0 1
),
and we recover the familiar representation for Ga (see Example 1.1.2(4)).
Remark 5.4.8. In general, the matrix group that we get by applying thisprocedure has the opposite multiplication compared to the group G(R). Seealso footnote 4 on page 58.
62
Chapter 6 Jordan decomposition
Now that we have shown that every affine algebraic group is linear, we canapply linear algebra to our study of linear algebraic groups. The Jordandecomposition in linear algebraic groups will allow us to decompose the el-ements in a semisimple and a unipotent part, and will have far-reachingconsequences. We begin with the study of this decomposition in the classicalsetting, namely in the matrix group GL(V ); we will then see how to extendour ideas to general linear algebraic groups.
6.1 Jordan decomposition in GL(V )
Definition 6.1.1. Let k be an arbitrary commutative field, let V be a finite-dimensional vector space over k, and let g ∈ Endk(V ). Then:
1. g is diagonalizable if V has a basis of eigenvectors for g;
2. g is semisimple if V has a basis of eigenvectors for g over the algebraicclosure k, i.e. if g is diagonalizable over k;
3. g is nilpotent if gN = 0 for some integer N ;
4. g is unipotent if g − 1 is nilpotent.
We can now state the main theorem of this section. Recall that a commu-tative field is called perfect if every irreducible polynomial over k has distinctroots, or equivalently, if either char(k) = 0, or char(k) = p and every elementof k is a p-th power.
Theorem 6.1.2 (Jordan decomposition in GL(V )). Let k be a perfect com-mutative field, let V be a finite-dimensional vector space over k, and letg ∈ GL(V ). Then there exist unique elements gs, gu ∈ GL(V ) such that:
(a) gs is semisimple;
(b) gu is unipotent;
(c) g = gsgu = gugs.
63
Moreover, both gs and gu can be expressed as polynomials in g without con-stant term.
Proof. We will prove the theorem for algebraically closed fields k only; theproof for general perfect fields k can either go along the same lines, or onecan alternatively reduce the general case to the case of algebraically closedfields by some general arguments.
We will first show existence of the elements gs and gu. Choose a basis forV such that g is in its Jordan normal form
g =
λ1 ∗λ1 ∗
. . . ∗λ1
. . .
λm ∗λm ∗
. . . ∗λm
,
where each ∗ is either 0 or 1, and where λ1, . . . , λm are the distinct eigenvaluesof g. Notice that each λi is non-zero since g is invertible. For each i ∈{1, . . . ,m}, we let Vi be the generalized eigenspace corresponding to λi; thedecomposition V = V1⊕· · ·⊕Vm corresponds to the lines in the above matrixfor g. Let
gs :=
λ1
λ1
. . .
λ1
. . .
λmλm
. . .
λm
,
and let gu := g−1s g = gg−1
s . Then gs and gu satisfy the requirements (a)–(c).
We now show uniqueness. So assume that g = hshu = huhs, where hsis semisimple and hu is unipotent. Let v be an arbitrary eigenvector for hs,
64
with eigenvalue λ. Then
(g − λ)N(v) = (huhs − λ)N(v) = (huhs − λ)N−1(hu − 1)λv
= λ(hu − 1)(huhs − λ)N−1(v) = · · · = λN(hu − 1)N(v);
since hu is unipotent, it follows that (g − λ)N(v) = 0 for large enough N .Hence v ∈ Vi for some i, and in particular λ = λi. It follows that thedecomposition of V into eigenspaces for the semisimple element hs coincideswith the decomposition V = V1 ⊕ · · · ⊕ Vm, with the same eigenvalues, andhence hs = gs. This implies hu = gu, showing that the Jordan decompositionof g is unique.
We finally show that gs and gu can be expressed as polynomials in g. Letni := dimVi. We apply the Chinese Remainder Theorem in k[x] to get apolynomial P (x) such that
P (x) ≡ λi mod (x− λi)ni for all i ∈ {1, . . . ,m};P (x) ≡ 0 mod x.
Then for each i ∈ {1, . . . ,m}, we have P (g)(v) = λiv for all v ∈ Vi, andhence P (g) coincides with gs. The condition P (x) ≡ 0 mod x ensures thatP (g) is a polynomial in g without constant term. Finally, observe that g−1
s isa polynomial in gs (consider its minimal polynomial), and hence gu = gg−1
s
is also a polynomial in g without constant term. �
Remark 6.1.3. The theorem does not hold when k is not perfect. Forinstance, let k be a field with char(k) = 2 such that there is some a ∈ k \ k2.Then
M =(
0 1a 0
)=(√
a 0
0√a
)(0 1/
√a√
a 0
),
so if M would have a Jordan decomposition over k, then it would haveat least two different Jordan decompositions over the algebraic closure k,contradicting the uniqueness.
Jordan decompositions behave well with respect to linear transformations.
Corollary 6.1.4. (i) Let V,W be finite-dimensional vector spaces over someperfect field k, and let ϕ : V → W be a linear transformation. Assumethat g ∈ GL(V ) and h ∈ GL(W ) are such that ϕ ◦ g = h ◦ ϕ. Then
ϕ ◦ gs = hs ◦ ϕ and ϕ ◦ gu = hu ◦ ϕ.
(ii) If U ≤ V is a g-invariant subspace, then it is also gs- and gu-invariant,and we have (g|U)s = (gs)|U and (g|U)u = (gu)|U .
65
Proof. Statement (i) follows from the fact that ϕmaps generalized eigenspacesto generalized eigenspaces: if (g − λ)N(v) = 0, then also
(h− λ)N(ϕ(v)
)= ϕ
((g − λ)N(v)
)= 0.
It follows that ϕ ◦ gs and hs ◦ϕ are identical on each generalized eigenspace,and hence on all of V . The same then holds for the unipotent part.
To prove (ii), consider the inclusion map ϕ : U → V , and apply (i). �
Although we are eventually interested in linear algebraic groups, whichwe know can be embedded in finite-dimensional matrix groups, we will haveto consider the more general case of infinite-dimensional vector spaces first.
Definition 6.1.5. Let k be an arbitrary field, and let V be an arbitraryk-vector space, possibly infinite-dimensional. Let g ∈ Endk(V ). Then:
(i) g is diagonalizable if V has a basis of eigenvectors for g;
(ii) g is semisimple if g is diagonalizable over k;
(iii) g is nilpotent if for each v ∈ V , there is some positive integer N suchthat gN(v) = 0;
(iv) g is unipotent if g − 1 is nilpotent;
(v) g is locally finite if for each v ∈ V , the subspace 〈gn(v) | n ∈ Z≥0〉 isfinite-dimensional.
Notice that unipotent elements and semisimple elements are locally finite.
Example 6.1.6. Let V = k[t], and consider the linear transformation D =ddt
, formal derivation in the variable t. Then D is locally finite and nilpotent.On the other hand, there is no positive integer N such that DN = 0.
Remark 6.1.7. If g ∈ GL(V ), then g is locally finite if and only if thesubspace 〈gn(v) | n ∈ Z〉 is finite-dimensional for each v ∈ V . In otherwords, if g ∈ GL(V ) is locally finite, then so is g−1. (Indeed, notice that g−1
is a polynomial in g.)
The following proposition shows that Jordan decomposition continues tohold for locally finite automorphisms.
Proposition 6.1.8. Let k be a perfect field, and let V be an arbitrary k-vectorspace, possibly infinite-dimensional. Let g ∈ GL(V ) be locally finite. Thenthere exist unique elements gs, gu ∈ GL(V ) such that:
(a) gs is semisimple;
66
(b) gu is unipotent;
(c) g = gsgu = gugs.
Moreover, every g-invariant subspace of V is also gs- and gu-invariant.
Proof. For every v ∈ V , we let
Lv := 〈gn(v) | n ∈ Z≥0〉.
Then each Lv is finite-dimensional, and g ∈ GL(V ) implies that the restrictiong|Lv belongs to GL(Lv). Hence we can apply Jordan decomposition to eachLv to obtain
g|Lv = (g|Lv)s · (g|Lv)u ;
this allows us to define
gs(v) := (g|Lv)s(v) and gu(v) := (g|Lv)u(v)
for each v ∈ V . Linearity of gs and gu follows by restricting to the finite-dimensional g-invariant subspaces containing the relevant1 elements of V ,together with Corollary 6.1.4(ii).
It is clear that gu is unipotent, because this is a local property. Onthe other hand, it is somewhat more subtle to see that gs is semisimple;this follows from the fact that V is a direct limit of its g-invariant finite-dimensional subspaces. �
6.2 Jordan decomposition in linear algebraic
groups
We now assume that G is a linear algebraic k-group. Our goal is to show thatevery element of G(k) admits a (canonical) Jordan decomposition when k isperfect. To obtain this goal, we will first study the regular representation forG, and afterwards we will move on to the finite-dimensional representations,which we are interested in.
Lemma 6.2.1. Let G be a linear algebraic k-group with coordinate algebraA = k[G], and consider its regular G-representation (A,∆), with correspond-ing action
ρ = ρk : G(k)→ GLA(k) = GL(A)
1For instance, if we want to show gs(v + w) = gs(v) + gs(w), then we consider thesmallest g-invariant subspace containing v, w and v + w, which is the finite-dimensionalsubspace 〈Lv, Lw, Lv+w〉, and we apply Jordan decomposition for the restriction of g tothat subspace, together with Corollary 6.1.4(ii).
67
given by equation (5.1). Then for every g ∈ G(k), the automorphism ρ(g) islocally finite.
Proof. Let v ∈ A be arbitrary. By Lemma 5.4.5, the representation (A,∆) islocally finite in the sense of Definition 5.4.4(ii), and hence v is contained insome finite-dimensional subrepresentationW ≤ V ; this means that ρ(g)(w) ∈W for all g ∈ G(k) and all w ∈ W . In particular, 〈ρ(g)n(v) | n ∈ Z≥0〉 iscontained in W , for all g ∈ G(k). �
This allows us to transfer our earlier definitions to the context of linearalgebraic groups:
Definition 6.2.2. Let G be a linear algebraic group defined over k, andconsider its regular G-representation (k[G],∆), with corresponding actionρ : G(k)→ GL(k[G]). Let g ∈ G(k).
(i) We call g semisimple if ρ(g) is semisimple.
(ii) We call g unipotent if ρ(g) is unipotent.
(iii) If g = gsgu = gugs with gs semisimple and gu unipotent, then we callthe pair (gs, gu) the Jordan decomposition of g.
Remark 6.2.3. (i) If g has a Jordan decomposition, then it is necessarilyunique because of Proposition 6.1.8; recall that ρ is injective becausethe regular representation is faithful.
(ii) Every ρ(g) has a Jordan decomposition in GL(k[G]), but it is not ob-vious at all that the corresponding elements ρ(g)s and ρ(g)u arise fromelements of G(k), i.e. whether they are contained in the image of ρ.
(iii) If ϕ : G → H is a morphism of linear algebraic groups, then ϕ mapssemisimple elements to semisimple elements and unipotent elements tounipotent elements. To see this, notice that if g ∈ G, then ρ(g) andρ(ϕ(g)) are related by the commutative diagram
k[H] k[G]
k[H] k[G]
ϕ∗
ρ(ϕ(g)) ρ(g)
ϕ∗
and apply an argument similar to Corollary 6.1.4 but now for locallyfinite automorphisms. In particular, ϕ preserves the Jordan decompo-sition of elements (assuming that it exists).
68
In order to proceed, we first need a connection between semisimple andunipotent elements of GL(V ) on the one hand, and of GL(k[GLV ]) on theother hand. We will only sketch the proof of this result since it is ratherspecific and has some technical details that we will not need again.
Lemma 6.2.4. Let V be a finite-dimensional vector space over k, and letG be the linear algebraic group G = GLV . Let g ∈ G(k) = GL(V ). Theng is semisimple (unipotent) if and only if ρ(g) ∈ GL(k[G]) is semisimple(unipotent).
In particular, if g ∈ G(k) has a Jordan decomposition (gs, gu), thenρ(g)s = ρ(gs) and ρ(g)u = ρ(gu).
Sketch of proof. The proof proceeds in three steps:
(1) ρ(g) is semisimple (unipotent) on k[GLV ] if and only if ρ(g) is semisimple(unipotent) on k[End(V )];
(2) k[End(V )] ∼= k[x11, . . . , xnn] ∼= Sym(End(V )∗), where
Sym(Z) :=∞⊕m=0
Z⊗m/〈x⊗ y − y ⊗ x | x, y ∈ Z〉.
Then ρ(g) is semisimple (unipotent) on k[End(V )] if and only if ρ(g) issemisimple (unipotent) on End(V )∗;
(3) ρ(g) is semisimple (unipotent) on End(V )∗ if and only if g is semisimple(unipotent) on V .
We refer, for instance, to [Sza12] for more details. �
We are now ready to state the Jordan decomposition for linear algebraicgroups.
Theorem 6.2.5 (Jordan decomposition in linear algebraic groups). Let G bea linear algebraic group defined over some perfect field k, and let g ∈ G(k).Then g has a unique Jordan decomposition (gs, gu). Moreover, for everyembedding ϕ : G ↪→ GLn, we have ϕ(gs) = ϕ(g)s and ϕ(gu) = ϕ(g)u.
Proof. Choose an arbitrary embedding ϕ : G ↪→ GLV with dimk V <∞, andlet A = k[GLV ]. Consider the corresponding dual morphism
ϕ∗ : A→ k[G];
by Corollary 5.3.3, ϕ∗ is surjective. Let I = kerϕ∗. Notice that an elementh ∈ GLV (k) ' homk-alg(A, k) belongs toG(k) if and only if h(I) = 0. (Indeed,
69
G(k) ' homk-alg(A/I, k), and hence the embedding ϕk : G(k) ↪→ GLV (k) isgiven explicitly by
ϕk : homk-alg(A/I, k)→ homk-alg(A, k) : f 7→ f ◦ ϕ∗.)
Next, we claim that for each h ∈ GLV (k), we have
h = ε ◦ ρ(h), (6.1)
where ρ = ρk : GLV (k)→ GL(A) is the regular representation of GLV on thek-points given by equation (5.1) applied to the comodule (A,∆), and whereε : A→ k is the counit of the Hopf algebra A. Indeed,
ε ◦ ρ(h) = ε ◦m ◦ (h⊗ idA) ◦∆
= m ◦ (idk ⊗ ε) ◦ (h⊗ idA) ◦∆
= m ◦ (h⊗ idk) ◦ (idA ⊗ ε) ◦∆
= h ◦m ◦ (idA ⊗ ε) ◦∆
= h.
We now claim thath(I) = 0 ⇐⇒ ρ(h)(I) ⊆ I. (6.2)
It is clear from (6.1) that if ρ(h)(I) ⊆ I, then h(I) = ε(ρ(h)(I)) ⊆ ε(I) = 0because I is a Hopf ideal. Conversely, if h(I) = 0, i.e., if h ∈ G(k), thenthe regular representations of GLV and of G are compatible for h, i.e. thefollowing diagram commutes:
ρ(h) : A A⊗ A k ⊗ A A
A/I A/I ⊗ A/I k ⊗ A/I A/I
∆
ϕ∗
h⊗ idA
ϕ∗ ⊗ ϕ∗
∼=
id⊗ ϕ∗ ϕ∗
∆G h⊗ idA/I ∼=
Therefore, ρ(h)(I) ⊆ I. (Alternatively, this can be deduced from the factthat ∆(I) ⊆ A⊗ I + I ⊗ A.) This proves the claim (6.2).
Each element g ∈ GLV (k) has a Jordan decomposition (gs, gu), so we onlyhave to show that g ∈ G(k) implies gs, gu ∈ G(k) as well, or equivalently,that
g(I) = 0 =⇒ gs(I) = gu(I) = 0. (6.3)
By Lemma 6.2.4, we have
ρ(g)s = ρ(gs) and ρ(g)u = ρ(gu).
70
If g(I) = 0, then by (6.2), the subspace I of A is ρ(g)-invariant. Because everyρ(g)-invariant subspace is also ρ(g)s- and ρ(g)u-invariant (Corollary 6.1.4(ii)),it follows that ρ(gs)(I) ⊆ I and ρ(gu)(I) ⊆ I. Again invoking (6.2), weconclude that gs(I) = gu(I) = 0 as claimed. Moreover, it follows from theuniqueness of the Jordan decomposition that
ϕ(gs) = ϕ(g)s and ϕ(gu) = ϕ(g)u
for each embedding ϕ : G ↪→ GLV . �
We end this chapter by mentioning an important consequence of the Jor-dan decomposition for commutative linear algebraic groups.
Definition 6.2.6. Let G be a linear algebraic group defined over some alge-braically closed field k. Then we define
Gs := {g ∈ G(k) | g is semisimple};Gu := {g ∈ G(k) | g is unipotent}.
Observe that Gu is always a closed subset of G, since after embedding itinto some GLn, it is determined by the polynomial equation (g−1)n = 0. Onthe other hand, the set Gs is not a closed subset in general. For commutativegroups however, the situation is much nicer.
Theorem 6.2.7. Let G be a commutative linear algebraic group defined oversome algebraically closed field k. Then both Gs and Gu are closed subgroupsof G, and the product map
Gs ×Gu → G
is an isomorphism.
Proof omitted. �
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Chapter 7 Lie algebras and
linear algebraic groups
In this chapter, we will see how we can associate a Lie algebra to everylinear algebraic group; this algebra will arise as the tangent space of thecorresponding algebraic variety, equipped with additional structure arisingfrom the group structure of the linear algebraic group. We will soon see thatour general point of view, describing a linear algebraic group G as a functorfrom k-alg to Grp, is also very convenient for this purpose: we will be usingthe k[ε]-points of G, where k[ε] is the ring of dual numbers defined as k+ kεwith ε2 = 0.
The Lie algebra is a smaller object than the Hopf algebra, and frequentlyis easier to analyze, but it can give substantial information about G, espe-cially in characteristic zero.
At the end of this chapter, we will use the Lie algebra to introduce avery important canonical representation for G, the so-called adjoint repre-sentation. This representation will be crucial in Chapter 11 when we studyreductive groups.
7.1 Lie algebras
We begin by recalling what a Lie algebra is.
Definition 7.1.1. Let k be a commutative field. A Lie algebra over k is ak-vector space g, together with a map
[·, ·] : g× g→ g,
such that:
(a) [·, ·] is k-bilinear;
(b) [x, x] = 0 for all x ∈ g;
(c) the Jacobi identity[x, [y, z]
]+[y, [z, x]
]+[z, [x, y]
]= 0
holds for all x, y, z ∈ g.
73
The map [·, ·] is called the Lie bracket of g.
Observe that by property (b), the Lie bracket is skew symmetric, i.e. forall x, y ∈ g, we have [x, y] = −[y, x].
Definition 7.1.2. Let g, g′ be Lie algebras.
(i) A Lie algebra morphism from g to g′ is a k-linear map α : g → g′
preserving the Lie bracket, i.e. such that α([x, y]) = [α(x), α(y)] for allx, y ∈ g.
(ii) A Lie subalgebra of g is a k-subspace h ≤ g such that [x, y] ∈ h for allx, y ∈ h, i.e. such that [h, h] ⊆ h.
(iii) An ideal of g is a k-subspace i ≤ g such that [g, i] ⊆ i. It is called aproper ideal if it is not equal to g itself.
(iv) The dimension of g is simply defined to be the dimension of the under-lying vector space.
Definition 7.1.3. Let A be an associative but not necessarily commutativek-algebra. Then we can associate a Lie algebra a to A, by declaring a = Aas a k-vector space, and [x, y] = xy − yx for all x, y ∈ A. We will denote aby Lie(A). It is straightforward to check that [x, x] = 0 for all x ∈ A andthat the Jacobi identity holds.
Example 7.1.4. Let A = Endk(V ) be the k-algebra of k-linear endomor-phisms of a vector space V . Then we denote the corresponding Lie algebraLie(A) by glV . In particular, if dimk V = n < ∞, then A ∼= Matn(k),and the corresponding Lie algebra will be denoted by gln. If we denote byEij ∈ Matn(k) the matrix with a 1 on the (i, j)-th position, and a 0 every-where else, then the Lie bracket satisfies the rule
[Eij, Epq] = δpjEiq − δiqEpjfor all i, j, p, q.
We will need one more construction of Lie algebras, namely the Lie alge-bra of derivations of a k-algebra.
Definition 7.1.5. Let A be a not necessarily commutative nor associativek-algebra.
(i) A k-derivation (or simply derivation) on A is a k-linear map D : A→ Asuch that the Leibniz rule
D(a · b) = D(a) · b+ a ·D(b)
holds, for all a, b ∈ A.
74
(ii) We denote the set of all k-derivations on A by Derk(A) or by Der(A).Notice that Der(A) is a k-subspace of Endk(A); as we will see in Propo-sition 7.1.6 below, Der(A) is in fact a Lie subalgebra of glA.
Notice that the composition of two derivations is not necessarily a deriva-tion again; we have
(D1 ◦D2)(a · b) = (D1 ◦D2)(a) · b+ a · (D1 ◦D2)(b)
+D1(a)D2(b) +D2(a)D1(b) (7.1)
for all D1, D2 ∈ Der(A) and all a, b ∈ A. However:
Proposition 7.1.6. Let A be a not necessarily commutative nor associativek-algebra. Then Der(A) is a Lie subalgebra of glA.
Proof. It follows immediately from equation (7.1) that for all D1, D2 ∈Der(A), the map [D1, D2] = D1 ◦ D2 − D2 ◦ D1 satisfies the Leibniz rule,and hence belongs to Der(A) again. �
When A is itself a Lie algebra g, an important class of derivations are theso-called inner derivations:
Definition 7.1.7. Let g be a Lie algebra. Then for each x ∈ g, we define
adg x : g→ g : y 7→ [x, y]
for all y ∈ g; we call this the inner derivation induced by x, or the adjointlinear map of x.
Proposition 7.1.8. Let g be a Lie algebra. Then:
(i) For each x ∈ g, the inner derivation adg x is a derivation of g, where weconsider g as a k-algebra with multiplication given by the Lie bracket.
(ii) Let ad(g) := {adg x | x ∈ g}. Then ad(g) is an ideal of the Lie algebraDer(g).
(iii) The mapadg : g→ Der(g) : x 7→ adg x
is a Lie algebra homomorphism.
Proof. (i) We have to check that for all x, y, z ∈ g, the identity
adg x([y, z]
)=[
adg x(y), z]
+[y, adg x(z)
]holds. This identity is equivalent to the Jacobi identity.
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(ii) This follows from the fact that
[adg x,D] = adg(−Dx)
for all D ∈ Der(g) and all x ∈ g.
(iii) It follows from the Jacobi identity again that
adg[x, y](z) = (adg x)(adg y)(z)− (adg y)(adg x)(z)
for all x, y, z ∈ g. �
Definition 7.1.9. Let g be a Lie algebra. The kernel of the map adg is calledthe center of g and denoted by Z(g); observe that
Z(g) = {x ∈ g | [x, g] = 0}.
7.2 The Lie algebra of a linear algebraic group
We will now explain how we can associate a Lie algebra to a linear algebraicgroup G. We will first define the underlying vector space, and afterwards wewill make clear how to define the Lie bracket.
Definition 7.2.1. Let R be a commutative ring with 1. Then we define1
the ring of dual numbers over R to be
R[ε] := R[x]/(x2) = R⊕ εR
with ε2 = 0. We will denote the canonical projection on the first componentby π, i.e.
π : R[ε]→ R : a+ εb 7→ a;
note that π is a ring homomorphism.
Notice that an element a + εb ∈ R[ε] is invertible if and only if a isinvertible in R; in this case, the inverse is given by
(a+ εb)−1 = a−1 − εa−2b.
1Note the subtle difference in notation: we use ε for the dual numbers and ε for thecounit of the Hopf algebra. This should not cause too much confusion, since the former isa ring element whereas the latter is a ring morphism.
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Definition 7.2.2. Let G be a linear algebraic k-group. For each R ∈ k-alg,we define
LieR(G) := ker(G(R[ε]
) G(π)−−−→ G(R)).
The Lie algebra of G is now defined as
Lie(G) := Liek(G) = ker(G(k[ε]) G(π)−−−→ G(k)
).
Notice that this definition only gives Lie(G) the structure of a group; it isnot obvious that Lie(G) can be made into a k-vector space (it is not evenobvious that it is an abelian group).
We will first have a look at the mother of all linear algebraic groups, GLn.
Example 7.2.3. Consider the linear algebraic group G = GLn. Then2
G(k[ε])
= {A+ εB | A ∈ GLn(k), B ∈ Matn(k)};
the inverse of an element A+ εB ∈ G(k[ε])
is given by
(A+ εB)−1 = A−1 − εA−1BA−1.
Hence
Lie(G) = {In + εB | B ∈ Matn(k)},
and the map
E : Matn(k)→ Lie(G) : B 7→ In + εB
is a bijection; notice that E(A)E(B) = E(A+B). In particular, we see thatLie(G) is indeed an abelian group. Observe that this law tells us that themap E behaves, in some sense, as an exponential map.
It is now clear that it makes sense to make Lie(GLn) into a Lie algebra,since Matn(k) has a natural Lie algebra structure, namely the Lie algebragln introduced in Example 7.1.4.
Since every linear algebraic group can be embedded into some GLn, wecan use this primary example to define the Lie algebra structure on Lie(G)for any linear algebraic group G.
2Recall that for each R ∈ k-alg, the set of R-points G(R) is given as the set of solutions
of the polynomial equation d · det(tij)− 1 = 0 in Rn2+1. When R = k[ε], write each tij as
aij+εbij and d = r+εs, and expand to get the above description for G(k[ε]). Alternatively,simply express that a matrix A+ εB ∈ Matn(R) is invertible.
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Definition 7.2.4. Let G be a linear algebraic k-group, and let Lie(G) be asin Definition 7.2.2. Choose an arbitrary embedding G ↪→ GLn, and noticethat this induces an embedding of Lie(G) as a subgroup of Lie(GLn) = gln.It turns out that Lie(G) is, in fact, a Lie subalgebra of Lie(GLn) = gln, andthat the Lie algebra Lie(G) is independent (up to isomorphism) of the chosenembedding G ↪→ GLn.
It is a natural question whether it is possible to give a more intrinsicdefinition of the Lie algebra Lie(G), which does not depend on an embeddingG ↪→ GLn. This is indeed possible. We state the result without proof.
Definition 7.2.5. Let G be a linear algebraic group defined over k, and letA = k[G] be its coordinate algebra. Then a k-derivation D ∈ Derk(A) iscalled left-invariant if
∆ ◦D = (id⊗D) ◦∆.
We will denote the space of left-invariant k-derivations on A by Der`k(A).
The space of left-invariant k-derivations is a Lie subalgebra of Derk(A):
Lemma 7.2.6. Let G be a linear algebraic group defined over k, and letA = k[G] be its coordinate algebra. Then Der`k(A) is a Lie subalgebra ofDerk(A).
Proof. We have to check that when D1, D2 ∈ Derk(A) are left-invariant, thenso is [D1, D2] = D1 ◦D2 −D2 ◦D1. This is an easy exercise. �
Theorem 7.2.7. Let G be a linear algebraic group defined over k, and letA = k[G] be its coordinate algebra. Then
Lie(G) ' Der`k(A)
as Lie algebras.
Proof omitted. �
We will now give some more examples; we leave some of the details tothe reader.
Example 7.2.8. (1) Let G = SLn. Then
Lie(G) = {In + εA ∈ Matn(k[ε]) | det(In + εA) = 1}.
Since ε2 = 0, we have det(In + εA) = 1 + ε tr(A), and hence
Lie(SLn) = {In + εA | A ∈ Matn(k), tr(A) = 0}= {E(A) | A ∈ Matn(k), tr(A) = 0}.
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We observe that Lie(SLn) is indeed a Lie subalgebra of gln, which wedenote by sln. In fact, we even have that the Lie bracket of any twoelements of gln belongs to sln, since for all A,B ∈ Matn(k), we havetr(AB −BA) = 0.
(2) Let G = Tn be the linear algebraic group of invertible upper triangularmatrices. Then Lie(G) is isomorphic to the Lie subalgebra of (all) uppertriangular matrices
Lie(Tn) = {E(A) | A ∈ Matn(k), Aij = 0 for all i > j}.
(3) Let G = Un be the linear algebraic group of upper triangular matriceswith 1’s on the diagonal. Then Lie(G) is isomorphic to the Lie subalgebra
Lie(Un) = {E(A) | A ∈ Matn(k), Aij = 0 for all i ≥ j}.
(4) Let G = Dn be the linear algebraic group of invertible diagonal matrices.Then Lie(G) is isomorphic to the Lie subalgebra of (all) diagonal matrices
Lie(Dn) = {E(A) | A ∈ Matn(k), Aij = 0 for all i 6= j}.
Remark 7.2.9. (i) A morphism of linear algebraic groups α : G → Hinduces a morphism of Lie algebras Lie(α) : Lie(G) → Lie(H), whichis injective if the morphism G → H is a closed embedding, i.e. if α isinjective and α(G) is a closed subgroup of H.The fact that we have a morphism of abelian groups from Lie(G) toLie(H) follows immediately from the definitions, and more preciselyfrom the commutative diagram
G(k[ε])
G(k)
H(k[ε])
H(k)
G(π)
αk[ε] αk
H(π)
but it requires more effort to show that this morphism Lie(α) preservesthe Lie brackets.
(ii) The Lie algebra construction is functorial. More precisely, the construc-tion described in (i) is the unique way of making Lie : G 7→ Lie(G) intoa functor (from linear algebraic k-groups to Lie k-algebras) such thatLie(GLn) = gln.
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One important feature of the Lie algebra of an algebraic group is that itprovides a natural representation, known as the adjoint representation. Todefine it, recall that
π : R[ε]→ R : a+ εb 7→ a,
and defineι : R→ R[ε] : a 7→ a+ ε0;
then π ◦ ι = idR. These maps give rise to homomorphisms
G(R)ι−−→ G(R[ε])
π−−→ G(R), π ◦ ι = idG(R),
where we have written π and ι instead of G(π) and G(ι) to simplify thenotation. Recall that
g(R) = ker(G(R[ε]
) π−−→ G(R)).
It is a not completely trivial fact that
g(R) ∼= g(k)⊗k R;
this follows most easily from the description of g(R) in terms of derivations,but we will omit the details. Now define
AdR : G(R)→ Aut(g(R)) : g 7→ AdR(g),
whereAdR(g) : g(R)→ g(R) : x 7→ ι(g) · x · ι(g)−1
for all g ∈ G(R). Notice that the map AdR(g) is in fact an R-linear map,and hence AdR maps G(R) into GL(g(R)). Since all the constructions arenatural in R, this gives rise to a natural transformation
Ad: G→ GLg.
Definition 7.2.10. Let G be a linear algebraic k-group. The adjoint repre-sentation of G is the representation Ad defined above.
As a useful example, we compute the adjoint representation for the groupG = GLn.
Example 7.2.11. Let k be a field and let G = GLn over k with Lie algebrag = gln(k). Let Ad: G→ GLg be the adjoint representation of G. Then theadjoint action of G = GLn on g is given by conjugation: Ad(A)(X) = AXA−1
for all A ∈ GLn(R) and all X ∈ g(R).
Indeed, the elements of g(R) are of the form I + εX for X ∈ Matn(R),and by definition, the adjoint action of an element A ∈ GLn(R) is given by
I + εX 7→ ι(A) · (I + εX) · ι(A)−1 = I + εAXA−1.
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The adjoint representation can be used to give another (but equivalent)definition of the Lie bracket on g = Lie(G):
Theorem 7.2.12. Let G be a linear algebraic k-group, with Lie algebra g,and with adjoint representation Ad: G→ GLg. Let ad be the adjoint map ofthe Lie algebra g as in Definition 7.1.7. Then Lie(Ad) = ad.
Proof. By Definition 7.2.4, it suffices to show this for G = GLn. The Liealgebra g = gln comes equipped with the adjoint map
ad: g→ glg : A 7→ ad(A),
where ad(A) acts on g as X 7→ [A,X] = AX −XA.
On the other hand, if we apply the functor Lie to the homomorphism Ad,we obtain a linear map
Lie(Ad): Lie(G)→ Lie(GLg) ∼= glg.
We already know from Example 7.2.11 that A ∈ GLn(R) acts on Matn(R)by mapping each X to AXA−1, so when we apply the Lie functor, we obtainthat an element I + εA ∈ Lie(GLn(R)) acts on Matn(R[ε]) by mapping eachX + εY to
(I + εA)(X + εY )(I + εA)−1 = X + εY + ε(AX −XA).
In other words, Lie(Ad)(A) acts as id + ε ad(A), as required. �
Remark 7.2.13. The adjoint representation is not faithful in general. No-tice, for instance, that Z(G) is always contained in the kernel of Ad. In fact,when char(k) = 0 and G is connected, then Z(G) = ker(Ad), but in general,this is not true. (If G is connected, then the quotient ker(Ad)/Z(G) is alwaysa unipotent group.)
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Chapter 8 Topological aspects
We will briefly study some of the topological aspects of linear algebraicgroups, and in particular we will study connectedness. This crucial propertywill have a nice interpretation in terms of the Hopf algebra coordinatizingthe linear algebraic group. At the end of this chapter, we will also say a fewwords about smoothness and the dimension of linear algebraic groups.
8.1 Connected components of matrix groups
Before we study connectedness for linear algebraic groups in general, it isenlightening to have a look at the “classical” case of closed1 subgroups ofGLn(k). Recall from Corollary 4.1.16 that every affine variety is a finite unionof its irreducible components (and in particular it is also a finite union of itsconnected components, each of which is a union of irreducible components).
Theorem 8.1.1. Let S be a closed subgroup of GLn(k), and let S◦ be the con-nected component containing the unit 1 ∈ S. Then S◦ is a normal subgroupof finite index in S. The irreducible components of S coincide with the con-nected components; they are precisely the cosets of S◦ in S, so in particularthere are precisely [S : S◦] components.
Proof. Let S = V1 ∪ · · · ∪ Vr be the decomposition of S into its irreduciblecomponents. Then V1 is not contained in any Vj (2 ≤ j ≤ r), and since V1 isirreducible, it is not contained in their union V2 ∪ · · · ∪Vr either; hence thereis some x ∈ V1 not contained in any other irreducible component. Since Sis a group, every left translation S → S : y 7→ gy (where g ∈ S is fixed)is a homeomorphism, and hence every element of S is contained in exactlyone irreducible component. It follows that the irreducible components aredisjoint, and hence they coincide with the connected components.
If x ∈ S◦, then the set xS◦ is homeomorphic to S◦ and hence a component;since it contains x ∈ S◦, this implies that xS◦ = S◦, and therefore S◦ is closedunder multiplication. For similar reasons S◦ is closed under inverses and is
1Of course, we are considering GLn(k) as an affine variety over k endowed with theZariski topology, as in Chapter 4.
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invariant under conjugation by any g ∈ S. Finally, since we have alreadyobserved that each coset xS◦ is an irreducible component, we have precisely[S : S◦] such components. �
8.2 The spectrum of a ring
Our next goal is to study connectedness for our more general notion of linearalgebraic groups as k-group functors. Such an object G is completely deter-mined by its coordinate algebra k[G], and we would like to see how we candetect connectedness in terms of this Hopf algebra.
But what does connectedness even mean for a k-functor? When k is alge-braically closed, it makes sense to consider the group of k-points G(k), andalgebraically the k-points are in one-to-one correspondence with the maxi-mal ideals (see Corollary 4.1.11). This is no longer true for general fields k,and the collection of maximal ideals does not capture enough information ingeneral, certainly not when we consider the group of R-points G(R) for somek-algebra R.
It turns out that considering all prime ideals instead is more satisfying,and that is what we will do.
Definition 8.2.1. Let A be a commutative ring with 1. Then the spectrumof A is defined as the collection of its prime ideals
SpecA := {I E A | I is prime}.
We make SpecA into a topological space by declaring a subset of SpecA tobe closed if it has the form
V (I) := {P ∈ SpecA | P ⊇ I}
for some ideal IEA. This topology2 is called the Zariski topology on SpecA.
To see the connection with the classical geometric objects, assume thatA = k[V ] for some affine variety V ⊆ kn over an algebraically closed field k.Then every point of V corresponds to a maximal ideal of A and hence to anelement of SpecA; this embedding V ↪→ SpecA induces a homeomorphismfrom V onto its image. Moreover, the image is dense: if a closed set V (I)contains V , then I is contained in the intersection of all maximal ideals, andsince A is noetherian, this intersection coincides with the intersection of all
2It is not hard to check that V (I) ∪ V (J) = V (IJ) and⋂V (Iα) = V
(∑Iα), so this
does indeed define a topology.
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prime ideals (see also the proof of Corollary 8.2.5(iii) below); this impliesthat indeed V (I) = SpecA. Recall that a topological space is irreducible ifand only if every non-empty open set is dense; it follows that V is irreducibleif and only if SpecA is irreducible. Also, if V is connected, then SpecAis also connected; the converse is also true, but this is less obvious (seeCorollary 8.2.5 below).
It is easy to detect irreducibility of SpecA from the structure of A. Com-pare this with Lemma 4.1.15.
Lemma 8.2.2. Let A be a commutative ring with 1, and let N be its nilrad-ical, i.e. the set of nilpotent elements of A. Then:
(i) N is an ideal; it coincides with the intersection of all prime ideals of A;
(ii) SpecA is irreducible if and only if A/N is an integral domain;
(iii) if A is noetherian, then SpecA is the union of finitely many maximalirreducible closed subsets, its irreducible components.
Proof. (i) Let a ∈ A be nilpotent, and P E A be prime. Then A/P is anintegral domain, hence the image of a in A/P is zero, and hence a ∈ P .Therefore every nilpotent element is contained in the intersection of allprime ideals.Conversely, let a ∈ A be non-nilpotent, and let Aa = A[a−1] be thelocalization of A at a. Take a maximal ideal I E Aa; its inverse imagein A is prime and does not contain a.
(ii) Assume first that SpecA = Y1 ∪ Y2 for some proper closed subsets Y1
and Y2. Then there exists an element a ∈(∩P∈Y1P
)\N , and an element
b ∈(∩P∈Y2P
)\N . Then each prime ideal P ∈ SpecA contains either a
or b (or both), and hence contains ab; so ab ∈ N . Since neither a ∈ Nnor b ∈ N , this shows that A/N is not an integral domain.Conversely, if A/N contains zero divisors, then we can find a, b ∈ Asuch that ab ∈ N but neither a ∈ N nor b ∈ N . Since ab ∈ N , we havefor each prime ideal P E A that either a ∈ P or b ∈ P , and it followsthat
SpecA = {P ∈ SpecA | a ∈ P} ∪ {P ∈ SpecA | b ∈ P}
decomposes SpecA as the union of two proper closed subspaces, henceSpecA is reducible.
(iii) Since A is noetherian, any non-empty collection of closed sets in SpecAhas a minimal element. We will show that all closed sets are finiteunions of irreducible closed subsets. Assume not; then by our previous
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observation, we can take a minimal closed set Y which is not a finiteunion of irreducible closed subsets. Then Y is certainly reducible, sayY = Y1 ∪ Y2; by minimality, Y1 and Y2 would be finite unions of irre-ducible closed subsets, but then the same would be true for Y itself,which is a contradiction. Hence we can write every closed X as a finiteirredundant union X = X1 ∪ · · · ∪Xr of irreducible closed subsets, andin particular this is true for SpecA itself. �
We now proceed to study connectedness of SpecA. An important role isplayed by the idempotents in A.
Theorem 8.2.3. Let A be a commutative ring with 1. A closed set V (I) inSpecA is clopen (i.e. closed and open) if and only if V (I) = V (e) for someidempotent element e ∈ A. Moreover, if V (e) = V (f) for some idempotentse, f ∈ A, then e = f .
Proof. Assume that e ∈ A is idempotent; then e + (1− e) = 1, so V (e) andV (1− e) are disjoint closed sets. On the other hand, if P EA is prime, then0 = e(1 − e) ∈ P implies e ∈ p or 1 − e ∈ P , and hence V (1 − e) is thecomplement of V (e), which implies that V (e) is clopen.
Assume that V (e) = V (f) for some idempotents e, f ∈ A. Then
V (f(1− e)) = V (f) ∪ V (1− e) = SpecA,
hence by Lemma 8.2.2(i), the element f(1−e) is nilpotent. However, f(1−e)is also idempotent, and hence f(1−e) = 0, implying f = ef . Similary e = efand we conclude that e = f .
Assume finally that V (I) is clopen, and write its complement as V (J).Then V (I + J) = V (I) ∩ V (J) = ∅ and hence I + J = A, which impliesthat we can write 1 = a + b with a ∈ I and b ∈ J . On the other hand,SpecA = V (I) ∪ V (J) = V (IJ), and hence ab is nilpotent, so we have(ab)N = 0 for some N . Notice that a maximal ideal containing aN and bN
would contain a and b and hence a+b = 1, which is a contradiction; hence wecan write 1 = uaN +vbN for some u, v ∈ A. Observe that uaN is idempotent,since
(uaN)2 = uaN · (1− vbN) = uaN − uv(ab)N = uaN .
On the other hand, we have
V (uaN) ⊇ V (I) and V (vbN) ⊇ V (J),
with V (uaN) disjoint from V (vbN); we conclude that V (I) = V (uaN). �
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Remark 8.2.4. Notice that if A is a ring with a non-trivial idempotent e,then A decomposes as the product
A ∼= eA× (1− e)A,
where eA and (1−e)A are rings with unit e and 1−e, respectively. Conversely,if A ∼= B × C for certain non-zero rings B and C, then A has non-trivialidempotents e = (1, 0) and 1− e = (0, 1).
Corollary 8.2.5. Let A be a commutative ring with 1.
(i) SpecA is connected if and only if A has no non-trivial idempotents.
(ii) If A is noetherian, then it has only finitely many idempotents.
(iii) If k is algebraically closed, and A is a finitely generated k-algebra, thenSpecA is connected if and only if its subset
SpecMaxA := {I E A | I is a maximal ideal}
is connected.
(iv) If k is algebraically closed, and V is an affine k-variety, then V isconnected if and only if Spec k[V ] is connected.
Proof. (i) This follows immediately from Theorem 8.2.3.
(ii) If A is noetherian, then by Lemma 8.2.2(iii), Spec(A) has only finitelymany irreducible components. Since every connected component is aunion of irreducible components, this implies that Spec(A) has onlyfinitely many connected components, and the result now follows againfrom Theorem 8.2.3.
(iii) The important point here is that Hilbert’s Nullstellensatz implies thatA is a Jacobson ring, i.e., every prime ideal is the intersection of themaximal ideals containing it; in particular, the nilradical N is equal tothe intersection of all maximal ideals of A. The proof of Theorem 8.2.3can now be adapted in order to get an idempotent element in A foreach clopen subset of SpecMaxA.
(iv) This follows from (iii) because V ∼= SpecMax k[V ] as topological spaces.�
8.3 Separable algebras
We now have a good definition of connectedness using the spectrum of thecoordinate algebra, but there is still an important problem that remains:
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the number of connected components is not always invariant under baseextension, i.e. extending the scalars of a k-algebra can create new idempotentelements. For example, consider the algebraic group of third roots of unity
µ3 : k-alg→ Grp : R 7→ {r ∈ R | r3 = 1},
with coordinate algebra
A = k[µ3] ∼= k[t]/(t3 − 1).
When k = R, SpecA has only two elements, and A has only two non-trivialidempotents, namely e = (t2 + t + 1)/3 and 1 − e; this corresponds to thefactorization t3 − 1 = (t− 1)(t2 + t+ 1). When k = C, however, SpecA hasthree elements, corresponding to the three roots of unity in C (and A hasmore idempotents). Also observe that over R, the two components are nothomeomorphic, in contrast to the situation we had in Theorem 8.1.1 above.
To resolve these issues, we will need a theory that detects these idempo-tents over base field extensions, and this is where separable algebras comeinto play.
Definition 8.3.1. Let k be a field, and let k be its algebraic closure. Acommutative k-algebra A is called separable if it is finite-dimensional andA⊗k k is reduced, i.e. does not have non-trivial nilpotent elements.
There exist several equivalent definitions; we mention just a few of thembelow, for later use.
Theorem 8.3.2. Let k be a field, let k be its algebraic closure, and let ks beits separable closure. Let A ∈ k-alg be finite-dimensional. Then the followingstatements are equivalent:
(a) A is separable;
(b) A⊗ k ∼= k × · · · × k;
(c) A⊗ ks ∼= ks × · · · × ks;(d) A is a product of separable extension fields of k;
(e) A⊗ k is reduced;
(f) (only when k is perfect:) A is reduced.
Proof omitted. �
Corollary 8.3.3. (i) Subalgebras, quotients, products and tensor productsof separable algebras are again separable.
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(ii) Let K/k be a field extension. Then A is separable over k if and only ifA⊗k K is separable over K.
Remark 8.3.4. (i) It can be shown that the category of separable k-al-gebras is anti-equivalent to the category of finite sets equipped witha continuous action of the absolute Galois group Gal(ks/k). This is asimple case of what is known as Galois descent: the classification overthe separable closure ks is easy, and the problem over arbitrary fieldsreduces to the study of k-forms, i.e. algebraic structures defined over kthat become isomorphic after a base change to the separable closure ks.
(ii) A finite linear algebraic k-group G is called etale 3 if k[G] is separable,and by (i), this corresponds to a finite set X on which Gal(ks/k) actscontinuously. In that case, the comultiplication ∆: k[G]→ k[G]⊗ k[G]gives a map X ×X → X commuting with the Galois action, and dual-izing brings this action back to a continuous action by group automor-phisms. Thus finite etale linear algebraic groups over k are equivalentto finite groups equipped with a continuous action of Gal(ks/k) by au-tomorphisms. If the Galois action on the finite group X is trivial, werecover the finite constant linear algebraic groups from Example 5.1.12,with A = kX .
We now go back to the situation where we have an affine k-functor withsome coordinate algebra A, which is a finitely generated k-algebra. The fol-lowing definition will be our essential tool to study connectedness in general.
Definition 8.3.5. Let A be a finitely generated k-algebra. Then there is aunique maximal separable subalgebra of A, which we denote by π0(A).
In order to see that π0(A) is unique, notice that if B is any separablesubalgebra, then its dimension is bounded by the number of connected com-ponents of SpecA⊗ k, since B⊗ k is also a separable k-subalgebra of A⊗ k,which is spanned by idempotents; moreover, if B and C are two separa-ble subalgebras of A, then so is the compositum BC, since it is a quotientof B ⊗ C.
The map A 7→ π0(A) behaves well with respect to product constructions:
Proposition 8.3.6. Let A and B be two finitely generated k-algebras, andlet L/k be a field extension. Then:
(i) π0(A×B) = π0(A)× π0(B);
(ii) π0(A⊗k L) ∼= π0(A)⊗k L;
3See also Definition 8.4.5(iii) below.
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(iii) π0(A⊗k B) ∼= π0(A)⊗k π0(B).
Proof. To see that (i) holds, note that π0(A× B) ⊆ π0(A)× π0(B) becausethe projections of π0(A×B) to A and B are separable, and π0(A)×π0(B) ⊆π0(A×B) because the product of two separable algebras is again separable.
The proof of (ii) and (iii) is more involved, and will be omitted. �
Remark 8.3.7. Let X be an affine k-functor, with A = k[X], and let π0(X)be the affine k-functor represented by the separable algebra π0(A). Thenwe can think of π0(X) as the functor describing the connected componentsof X. Notice that every idempotent e ∈ A is contained in π0(A) because k[e]is separable. There can be other nontrivial fields contained in π0(A), butsince
π0(A)⊗k k ∼= k × · · · × k,
these fields reflect potential idempotents, and hence components of X afterbase extension. By Proposition 8.3.6(ii), every such potential idempotent isindeed captured by π0.
8.4 Connected components of linear algebraic
groups
We are now fully prepared to study connectedness of linear algebraic groupsin general.
Theorem 8.4.1. Let G be a linear algebraic group defined over k, and letA = k[G] be its coordinate algebra. Then the following are equivalent:
(a) SpecA is connected;
(b) SpecA is irreducible;
(c) π0(A) = k;
(d) A modulo its nilradical is an integral domain.
Proof. Denote the nilradical of A by N . By Lemma 8.2.2(ii), (b) ⇐⇒ (d),and of course (b) =⇒ (a). Now assume that SpecA is connected; thenπ0(A) is a separable extension field of k. The counit ε : A→ k restricts to ak-algebra homomorphism π0(A) → k, which implies that π0(A) = k; hence(a) =⇒ (c). Conversely, assume that π0(A) = k. Then A does not havenon-trivial idempotents, so by Corollary 8.2.5(i), SpecA is connected; hence(c) =⇒ (a).
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We finally show that (c) =⇒ (d). So assume again that π0(A) = k; thenπ0(A ⊗ k) = k as well. In order to show that A/N is an integral domain,we may assume that k = k. In that case4, A/N is the ring of functions onthe group of k-points G(k). Since we have already shown that (c) =⇒ (a),we know that SpecA is connected; Corollary 8.2.5(iv) now implies that alsoG(k) is connected. By Theorem 8.1.1, we can now conclude that G(k) isirreducible, and hence its ring of functions A/N is an integral domain. �
Definition 8.4.2. If G is a linear algebraic group satisfying each of the fourequivalent conditions of Theorem 8.4.1, then we call G connected.
Corollary 8.4.3. Let G be a linear algebraic group defined over k, and letK/k be a field extension. Then G is connected if and only if GK is connected.
Proof. This follows from Proposition 8.3.6(ii) and condition (c) of Theo-rem 8.4.1. �
When G is not connected, the algebra π0(k[G]) is exactly what we needto analyze the connected components of G.
Proposition 8.4.4. Let G be a linear algebraic group defined over k, and letA = k[G] be its coordinate algebra. Then π0(A) is a Hopf subalgebra of A.
Proof. Notice that every k-algebra homomorphism f : A → B maps sepa-rable subalgebras onto separable subalgebras, and in particular f(π0(A)) ⊆π0(B). Since ∆: A→ A⊗ A and S : A→ A are k-algebra homomorphisms,we get
∆(π0(A)) ⊆ π0(A⊗ A) ∼= π0(A)⊗ π0(A) and S(π0(A)) ⊆ π0(A);
moreover, the counit ε : A→ k restricts to a k-algebra morphism π0(A)→ k.This shows that π0(A) is a Hopf subalgebra of A. �
Definition 8.4.5. Let G be a linear algebraic k-group, and let A = k[G].
(i) The linear algebraic group associated to the Hopf subalgebra π0(A)of A will be denoted by π0(G), and is called the group of connectedcomponents of G.
(ii) The kernel of the homomorphism G → π0(G) is called the identitycomponent of G, and is denoted by G◦.
(iii) The linear algebraic groupG is called etale if π0(A) = A, or equivalently,if G◦ is trivial.
4See also Remark 8.5.16 below.
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We list a few properties of π0(G) and G◦ in the next two propositions.
Proposition 8.4.6. Let G be a linear algebraic k-group, and let A = k[G],with counit ε : A → k. Then k[G◦] ∼= eA for some idempotent e ∈ A withε(e) = 1 and π0(eA) = k. In particular, G◦ is connected.
Proof. Consider the counit ε : π0(A)→ k, and use Theorem 8.3.2(d) to write
π0(A) = K1 × · · · ×Kr
where each Ki is a separable extension field of k. Since every idempotentin π0(A) is mapped to 0 or 1, there is exactly one Ki which is mapped to k(and which is therefore isomorphic to k), while all others are mapped to 0.Assume that this happens for i = 1, and let e = (1, 0, . . . , 0) ∈ K1×· · ·×Kr.Then we can write
A = eA× (1− e)A, (8.1)
and in particular π0(eA) = k and π0((1− e)A) = K2 × · · · ×Kr; notice thatthe latter is precisely the augmentation ideal I of π0(A), i.e. the kernel of thecounit ε : π0(A)→ k.
By Proposition 5.2.6, the kernel G◦ of the homomorphism G → π0(G)is a closed normal subgroup of G, with coordinate algebra k[G◦] ∼= A/IA.Explicitly, we have I = (1− e)π0(A), and hence, by (8.1),
k[G◦] ∼= A/(1− e)A ∼= eA.
It follows that π0(k[G◦]) ∼= π0(eA) = k, and hence G◦ is connected. �
Proposition 8.4.7. Let G be a linear algebraic k-group, and let A = k[G].
(i) Every homomorphism from G to an etale linear algebraic group H fac-tors uniquely through G→ π0(G).
(ii) Every homomorphism from a connected linear algebraic group to anetale linear algebraic group is trivial.
(iii) Every homomorphism from a connected linear algebraic group to G fac-tors through G◦ → G.
(iv) The functors G 7→ π0(G) and G 7→ G◦ commute with base field exten-sion.
(v) If G,H are linear algebraic k-groups, then π0(G×H) ∼= π0(G)×π0(H)and (G×H)◦ ∼= G◦ ×H◦.
Proof. (i) We have already observed that every morphism from a separablealgebra to A has its image in π0(A); the result follows by dualizing.
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(ii) Assume that G is connected, H is etale, and α : G→ H is a homomor-phism. By (i), α factors through G → π0(G). Since G is connected,however, π0(G) = 1, and hence α is trivial.
(iii) If H is connected, then the composition H → G → π0(G) has trivialimage, which shows that H → G factors through G◦ → G.
(iv) Since K[GK ] ∼= k[G] ⊗k K for every field extension K/k, this followsfrom Proposition 8.3.6(ii).
(v) Since k[G⊗H] ∼= k[G]⊗k k[H], this follows from Proposition 8.3.6(iii).�
Corollary 8.4.8. Let1→ N → G→ Q→ 1
be an exact sequence of linear algebraic k-groups. If N and Q are connected,then G is connected. Conversely, if G is connected, then Q is connected.
Proof. Assume first that N and Q are connected. Then N is contained inthe kernel of the map G→ π0(G), so by Proposition 5.3.7, this map factorsthrough G → Q, and so it induces an epimorphism from the connectedgroup Q to an etale group π0(G). By Proposition 8.4.7(ii), this epimorphismis trivial, which implies that π0(G) = 1, showing that G is connected.
Conversely, assume that G is connected, and consider the composition ofepimorphisms G → Q → π0(Q). Again, Proposition 8.4.7(ii) implies thatthis map is trivial, and hence π0(Q) = 1, showing that Q is connected. �
Example 8.4.9. (1) The linear algebraic groups Ga, GLn, Tn, Un, Dn areconnected because their coordinate algebra is an integral domain.
(2) Let G be the linear algebraic group of monomial matrices. Then π0(G)is the constant algebraic group Symn, and G◦ = Dn.
(3) The natural isomorphism (of affine k-functors, not of k-group functors!)
SLn(R)×Gm(R)→ GLn(R) : (A, r) 7→ A · diag(r, 1, . . . , 1)
defines an isomorphism of k-algebras
k[GLn] ∼= k[SLn]⊗k k[Gm] ∼= k[SLn]⊗k k[t, t−1],
and hence k[GLn] contains k[SLn] as a subring, which is therefore also anintegral domain; this shows that SLn is connected.
(4) Let k be a field of characteristic p, let n ≥ 2 be an integer, and considerthe algebraic group G = µn of n-th roots of unity over k. Recall thatA = k[µn] ∼= k[t]/(tn − 1).
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• If p 6= 0 and n is a power of p, then the nilradical of A is theideal N = (t−1); in this case, A/N ∼= k, and hence G is connected.(Geometrically, G consists of one “thick point” with multiplicity n.)
• Assume next that p = 0 or p > 0 and p - n. Then the nilradicalof A is trivial, and A is not an integral domain (it has zero divisort − 1); hence G is disconnected. (Geometrically, G consists of npoints, each with multiplicity 1.)
• Finally, assume that p > 0 and n = pr ·m with p - m and r ≥ 1.Then the nilradical of A is the ideal N = (tm − 1), but A/N isnot an integral domain (again, it has zero divisor t − 1); hence Gis disconnected. (Geometrically, G consists of m thick points, eachwith multiplicity pr.) Notice that in this case, µn ∼= µpr ×µm, so µnis the product of the connected group µpr and the etale group µm.
8.5 Dimension and smoothness
We will briefly mention some aspects of dimension and smoothness of linearalgebraic groups, without proofs.
Throughout this section, let k be an arbitrary field and let G be a linearalgebraic k-group with coordinate algebra A = k[G]. Denote the nilradicalof A by N .
We begin with the important and useful notion of dimension, which will,in particular, allow us later to prove certain statements by induction. Thereader should compare this with the earlier Definition 4.2.6.
Definition 8.5.1. (i) Assume first that G is connected. Then we definedimG := trdegk(Frac(A/N)), the transcendence degree over k of thefield of fractions of A/N . (Notice that A/N is an integral domain byTheorem 8.4.1.)
(ii) When G is not connected, we define dimG := dimG◦.
Remark 8.5.2. Equivalently, the dimension of G can be defined to be theKrull dimension of its coordinate ring A = k[G], i.e., the largest possibleheight of a maximal ideal in A (which is a finite number). (The height ht(p)of a prime ideal p is defined as the largest possible length n of a descendingchain
p = p0 ⊃ p1 ⊃ · · · ⊃ pn
of prime ideals in A.) Moreover, every maximal chain of distinct prime idealsin A/N has length dimG.
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Example 8.5.3. (i) A group G is zero-dimensional if and only if A/N hastranscendence degree 0, if and only A has Krull dimension zero, if andonly if A is finite-dimensional over k. (The last equivalence relies onthe Noether Normalization Lemma.) By definition, these are preciselythe finite linear algebraic groups.
(ii) Let G = SLn, A = k[G] ∼= k[t11, . . . , tnn] / (det(tij)−1). Notice that A isitself an integral domain, and hence dimG is equal to the transcendencedegree of Frac(A) over k, which is n2 − 1.
There is a close relation between the dimension of a linear algebraic groupand the dimension of its Lie algebra. They very often coincide, but notalways; this is precisely what gives rise to the notion of smoothness.
Proposition 8.5.4. Let G be a linear algebraic k-group, and let g = Lie(G).Then dimG ≤ dim g.
Definition 8.5.5. A linear algebraic group G is called smooth if dimG =dim Lie(G).
Example 8.5.6. Typical examples of non-smooth groups are the linear al-gebraic groups µp and αp over fields k of characteristic p. Compute as anexercise that these 0-dimensional groups have a 1-dimensional Lie algebra.
Remark 8.5.7. We have preferred to give the shortest and most direct def-inition of smoothness, but the notion also makes sense for algebraic varieties(as k-functors) in general. In that setting, an affine k-functor V with coor-dinate algebra A = k[V ] is called smooth if Vk is regular, i.e., if for everymaximal ideal m of A, the local ring Am is regular.
For perfect fields, we can see directly from the coordinate algebra whetherthe linear algebraic group is smooth.
Proposition 8.5.8. Let G be a linear algebraic k-group, where k is perfect,and let A = k[G]. Then G is smooth if and only if A is reduced, i.e. A doesnot contain non-zero nilpotent elements.
Example 8.5.9. Let k be a non-perfect field of characteristic p, and let a ∈ kbe an element that is not a p-th power. Then the subgroup G of Ga × Ga
defined by Y p = aXp is reduced but not smooth.
On the other hand, for fields of characteristic zero, the situation is verynice.
Theorem 8.5.10 (Cartier, 1962). Let G be a linear algebraic k-group, wherechar(k) = 0. Then G is smooth.
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Proposition 8.5.11. Quotients and extensions of smooth linear algebraicgroups are smooth.
Remark 8.5.12. The kernel of a homomorphism of smooth linear algebraicgroups need not be smooth. For example, in characteristic p, the kernelsof Gm → Gm : x 7→ xp and of Ga → Ga : x 7→ xp are precisely µp and αp,respectively, and these are not smooth.
The following useful results illustrate that dimensions behave nicer whenthe groups are smooth.
Proposition 8.5.13. Let G be a smooth linear algebraic k-group, and let Hbe a closed subgroup of G. Then the following are equivalent:
(a) dimH = dimG;
(b) H has finite index in G;
(c) G◦ = H◦.
Corollary 8.5.14. Let G be a smooth connected linear algebraic k-group,and let H be a proper closed subgroup of G. Then dimH < dimG.
Theorem 8.5.15. If 1→ N → G→ Q→ 1 is exact, then
dimG = dimN + dimQ.
Remark 8.5.16. From now on, we will often restrict to smooth linear alge-braic groups over an algebraically closed field k. In this case, the coordinatealgebra A = k[G] is reduced, and in fact, this means that it is the coordinatealgebra of an algebraic variety in the classical sense (see Proposition 4.2.3).In such cases, it is safe to identify G with its group of k-points G(k), and wewill often do so. We will freely make use of well-known notions and construc-tions from group theory, such as normal subgroups, normalizers, centralizers,etc., the construction and definition of which is far from obvious in the gen-eral setting, but which coincides with the classical notions applied on G(k)in the case of smooth linear algebraic groups over algebraically closed fields.
Of course, it is a restriction to only consider smooth linear algebraicgroups over an algebraically closed field, but on the other hand, this will giveus substantial information even for general linear algebraic groups. Indeed,if G is a linear algebraic group defined over an arbitrary field k, then we canbase change to the algebraic closure to get a group Gk which is defined over analgebraically closed field; and if Gk is not smooth, then we can “smoothen”it by replacing its coordinate algebra A = k[Gk] by A/N , where N is thenilradical of A, i.e. the ideal consisting of the nilpotent elements of A.
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Chapter 9 Tori and characters
An (algebraic) torus is a linear algebraic k-group that becomes isomorphicto Gm × · · · × Gm over the algebraic closure, and as such, tori are rathereasy to understand. However, we will see later that understanding the toriinside a given linear algebraic group G already reveals important aspects ofthe structure of G, and that is the main reason to study tori separately inthis chapter.
We will begin, however, by studying characters in linear algebraic groups;these are, in some sense, dual to subtori, and the two notions are interrelated.Characters will also play an important role in the representation theory oflinear algebraic groups.
9.1 Characters
Definition 9.1.1. Let G be a linear algebraic k-group, and let A = k[G].
(i) A character of G is a homomorphism χ : G → Gm, or equivalently, aHopf algebra homomorphism χ∗ : k[t, t−1]→ A.
(ii) Let X(G) be the set of all characters of G. We make X(G) into anabelian group by setting
(χ+ χ′)R(g) := χR(g) · χ′R(g) ∈ R×
for all χ, χ′ ∈ X(G), all R ∈ k-alg, and all g ∈ G(R); we call it thecharacter group of G.
(iii) If Γ is a finitely generated abelian group, then the group algebra kΓ isa finitely generated k-algebra (see Example 2.1.4(4)). If we define
∆(γ) := γ ⊗ γ, S(γ) := γ−1, ε(γ) := 1,
for all γ ∈ Γ, then kΓ becomes a Hopf algebra.
(iv) A non-zero element a ∈ A is group-like if ∆(a) = a⊗ a. Any group-likeelement a automatically satisfies S(a) = a−1 and ε(a) = 1.
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Lemma 9.1.2. Let G be a linear algebraic k-group, and let A = k[G]. Themap
α : X(G)→ {a ∈ A | a is group-like} : χ 7→ χ∗(t)
is a group isomorphism.
Proof. Notice that the set of group-like elements in A is indeed closed undermultiplication in A because ∆ is an algebra homomorphism. We first checkthat χ∗(t) is indeed group-like for every χ ∈ X(G). Indeed, we have ∆(t) =t ⊗ t in k[Gm] = k[t, t−1], and hence ∆(χ∗(t)) = χ∗(t) ⊗ χ∗(t) since χ∗ is aHopf algebra homomorphism.
Conversely, if a ∈ A is group-like, then we define
ψ : k[t, t−1]→ k[G] : t 7→ a.
Then (∆◦ψ)(t) =((ψ⊗ψ)◦∆
)(t), and this implies that ψ is a Hopf algebra
homomorphism, and hence ψ = χ∗ for some χ ∈ X(G). This already showsthat α is a bijection.
We finally check that α is a group homomorphism. Indeed, let χ1, χ2 ∈X(G). Then
(χ1+χ2)∗(t) = (χ1+χ2)(idA)(t) = χ1(idA)(t)·χ2(idA)(t) = χ∗1(t)·χ∗2(t). �
9.2 Diagonalizable groups
Before we move on to tori, we study the related class of diagonalizable groups.
Definition 9.2.1. Let G be a linear algebraic k-group, and let A = k[G].Then G is called diagonalizable if there is a Hopf algebra isomorphism A ∼= kΓfor some finitely generated abelian group Γ.
This definition might look surprising, and the connection with the clas-sical notion of diagonalizable matrix groups might be unclear at this point.This will become more transparant when we look at two examples.
Examples 9.2.2. (1) Let Γ = Z, and recall that kZ ∼= k[t, t−1]. (Notice thatthe isomorphism is indeed a Hopf algebra isomorphism.) We recognizethis as the coordinate algebra of the linear algebraic group G = Gm, i.e.Gm is diagonalizable.
(2) Let Γ = Z/nZ, and recall that k[Z/nZ] ∼= k[t]/(tn − 1). (Again, noticethat the isomorphism is indeed a Hopf algebra isomorphism.) We recog-nize this as the coordinate algebra of the linear algebraic group G = µn;see Example 5.1.2(6). Hence µn is diagonalizable.
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Recall that every finitely generated abelian group is a direct productof (finite or infinite) cyclic groups, which means that we have essentiallydiscovered all diagonalizable groups.
Theorem 9.2.3. Let G be a linear algebraic k-group. Then G is diagonal-izable if and only if it is isomorphic to a finite direct product of Gm’s andµn’s.
Proof. Observe that the group algebra k(Γ × Γ′) is isomorphic, as a Hopfalgebra, to kΓ⊗kkΓ′. The result now follows from the classification of finitelygenerated abelian groups together with Examples 9.2.2 and Remark 5.1.10.
�
Remark 9.2.4. This is one of the many instances where the functorial ap-proach to linear algebraic groups shows its advantages. In the classical set-ting, the corresponding result has an assumption on char(k), but this as-sumption is not needed here. Indeed, recall that when char(k) = p | n, thecoordinate algebra k[µn] is not reduced, and hence in this case µn does notarise from an affine variety in the classical sense; see Proposition 4.2.2.
Our next goal is to characterize diagonalizable groups by their characters,or equivalently, by their group-like elements. We first need a lemma.
Lemma 9.2.5. Let G be a linear algebraic k-group, and let A = k[G]. Thenthe group-like elements of A are linearly independent over k.
Proof. Exercise; use the fact that ∆ is an algebra homomorphism. �
Proposition 9.2.6. Let G be a linear algebraic k-group, and let A = k[G].Then G is diagonalizable if and only if A is spanned by its group-like ele-ments (as a vector space). Moreover, there is an anti-equivalence betweendiagonalizable groups and finitely generated abelian groups, given by
G↔ X(G).
Proof. Let Γ ⊆ A be the set of group-like elements in A, and recall fromLemma 9.1.2 that Γ is an abelian group, and that there is an isomorphismα : X(G)→ Γ.
Assume first that A = 〈Γ〉; by Lemma 9.2.5, this implies that Γ is a basisfor the k-vector space A. Hence α extends to a unique algebra isomorphismα : kX(G) → 〈Γ〉 = A, and in particular this shows that A ∼= kΓ. No-tice that this isomorphism is indeed a Hopf algebra isomorphism since thecomultiplication on the generating set Γ of both algebras coincides.
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Conversely, assume thatG is diagonalizable, withA ∼= kΓ for some finitelygenerated abelian group Γ. Then by definition, A = 〈Γ〉 as a vector space,and the elements of Γ are indeed group-like in kΓ.
Finally, notice that if ϕ : G→ H is a morphism of linear algebraic groups,then the dual map ϕ∗ : k[H] → k[G] maps group-like elements of k[H] togroup-like elements of k[G]. �
We now come to the important connection with representation theory.
Definition 9.2.7. Let G be a linear algebraic k-group.
(i) Let (V,m) be a G-representation. Then (V,m) is diagonalizable if itcan be decomposed as a sum of one-dimensional subrepresentations.(It is not hard to see that it can then be decomposed as a direct sumof one-dimensional subrepresentations.)
(ii) Each one-dimensional G-representation (L,m) defines a character χ,because GLL ∼= Gm.
(iii) Conversely, let χ : G → Gm be a character of G. Then χ defines arepresentation of G on any finite-dimensional vector space V by lettingg ∈ G(R) act on VR as multiplication by χ(g) ∈ R×.
(iv) Let χ be a character of G, and let (V,m) be any representation of G,with corresponding homomorphism ρ : G → GLV . We say that G actson V through χ if
ρ(g)(v) = χ(g)v
for all g ∈ G(R) and all v ∈ VR, or equivalently, if
m(v) = α(χ)⊗ v
for all v ∈ V . (Notice that g(α(χ)) = χ(g).)
(v) Let (V,m) be a G-representation. Then a non-zero v ∈ V is an eigen-vector for the representation, with corresponding character χ, if m(v) =α(χ)⊗ v.
(vi) Let (V,m) be a G-representation. Define Vχ to be the largest subspaceof V such that G acts on Vχ through the character χ:
Vχ :={v ∈ V | m(v) = α(χ)⊗ v
}.
If Vχ is non-trivial, we call it an eigenspace for G (or for the G-repre-sentation) with character χ.
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Theorem 9.2.8. Let G be a linear algebraic k-group, and let A = k[G]. ThenG is diagonalizable if and only if every representation of G is diagonalizable,if and only if every representation (V,m) of G has a decomposition
V =⊕
χ∈X(G)
Vχ. (9.1)
Proof. Assume first that G is diagonalizable, and let (V,m) be a G-repre-sentation. We have to show that V is spanned by elements u such thatm(u) ∈ A ⊗ ku. By Proposition 9.2.6, we know that A is spanned, as ak-vector space, by its subset Γ of group-like elements.
Now let v ∈ V be arbitrary, and write
m(v) =∑γ∈Γ
γ ⊗ uγ,
where each uγ ∈ V , and where the sum is a finite sum. We now apply thecomodule identities (see Definition 5.4.1(ii)) on v:
(idA ⊗m)(m(v)) = (∆⊗ idV )(m(v)) and
(ε⊗ idV )(m(v)) = idV (v)
yield ∑γ∈Γ
γ ⊗m(uγ) =∑γ∈Γ
γ ⊗ γ ⊗ uγ and (9.2)∑γ∈Γ
uγ = v, (9.3)
respectively. Since Γ is a basis of A, equation (9.2) shows that m(uγ) =γ ⊗ uγ ∈ A⊗ uγ for each γ ∈ Γ; equation (9.3) shows that v ∈ 〈uγ | γ ∈ Γ〉.This shows that V is spanned by elements u such that m(u) ∈ A⊗ ku.
Conversely, assume that every representation of G is diagonalizable. Thenin particular, the regular representation (V,m) = (A,∆) of G is diagonaliz-able, and hence A is spanned by its eigenvectors. Let a ∈ A be an eigenvectorfor the regular representation with character χ; then ∆(a) = m(a) = α(χ)⊗a.Applying the identity mult ◦ (id⊗ ε) ◦∆ = id on a yields
ε(a)α(χ) = a,
i.e. a is a scalar multiple of α(χ). It follows that A is spanned by its group-likeelements, i.e. G is diagonalizable.
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We finally show that a given G-representation (V,m) is diagonalizable ifand only if (9.1) holds. Clearly, every subspace Vχ is diagonalizable, hence(9.1) implies that (V,m) itself is diagonalizable. Conversely, assume that(V,m) is diagonalizable. Then V is spanned by eigenvectors, and since everyeigenvector belongs to some Vχ, we certainly have V =
∑χ∈X(G) Vχ. In
order to show that the sum is direct, assume that there exists a finite set ofcharacters χ1, . . . , χ` and corresponding non-zero eigenvectors v1, . . . , v` suchthat v1 + · · ·+ v` = 0. Applying m yields
α(χ1)⊗ v1 + · · ·+ α(χ`)⊗ v` = 0,
which contradicts Lemma 9.2.5. �
9.3 Tori
Definition 9.3.1. Let G be a linear algebraic k-group.
(i) The group G is a torus if Gk∼= (Gm)n for some integer n ≥ 1, where
k is the algebraic closure of k. Equivalently, G is a torus if and only ifGk is a smooth connected diagonalizable group.
(ii) The group G is called of multiplicative type if Gk is a diagonalizablegroup. In particular, every torus is of multiplicative type.
As we indicated, tori are especially useful when considered as subgroupsof a larger linear algebraic group. We will now show how we can associate afinite group to every such torus; this finite group will play an important rolelater when we describe the structure of reductive linear algebraic groups.
Theorem 9.3.2. Let G be a smooth linear algebraic group over an alge-braically closed field k, and let T be a torus contained in G.
(i) Let V be a (not necessarily faithful) finite-dimensional G-representa-tion. Let
M := {χ ∈ X(T ) | Vχ 6= 0};then M is a finite set. The normalizer1 NG(T ) permutes the subspaces{Vχ | χ ∈ M}, and hence induces an action of NG(T ) on M . Thekernel of this action contains CG(T ), and coincides with CG(T ) if therepresentation is faithful.
(ii) The group W (G, T ) := NG(T )/CG(T ) is finite.
1We define the normalizer and centralizer as concrete subgroups of G(k); see Re-mark 8.5.16.
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Proof. (i) By Theorem 9.2.8, V decomposes as V =⊕
χ∈X(T ) Vχ with re-spect to T . Since V is finite-dimensional, the set M is finite. Noticethat when we identify G with G(k) and V with V (k), the subspaces Vχcan be interpreted as k-subspaces
Vχ = {v ∈ V | t.v = χ(t)v for all t ∈ T},
where we have written t.v in place of ρ(t)(v).Assume now that g ∈ NG(T ), and define, for each character χ ∈ X(T ),a new character g.χ by
(g.χ)(t) := χ(g−1tg)
for all t ∈ T . We claim that g maps Vχ to Vg.χ. Indeed, let v ∈ Vχ bearbitrary; then for all t ∈ T ,
t.(g.v) = g.(g−1tg).v = g.χ(g−1tg)v = χ(g−1tg)g.v = (g.χ)(t) · g.v,(9.4)
showing that g.v ∈ Vg.χ indeed. Obviously, non-empty eigenspaces aremapped to non-empty eigenspaces, and hence NG(T ) acts on the finiteset M .We will now determine the kernel of this action. So let g ∈ NG(T ); theng is in the kernel of the action if and only if g.χ = χ for all χ ∈M . Byequation (9.4), this is equivalent to
t.g.v = χ(t)g.v (9.5)
for all t ∈ T , all χ ∈ M , and all v ∈ Vχ. Notice that χ(t) is a scalar,and χ(t)v = t.v since v ∈ Vχ; hence χ(t)g.v = g.t.v. It follows that(9.5) is in turn equivalent with
t.g.v = g.t.v
for all t ∈ T and all v ∈ Vχ, for all χ ∈ M . Since V is spanned by thesubspaces Vχ, this is equivalent with saying that the commutator [g, t]acts trivially on V , for all t ∈ T . In particular, CG(T ) is contained inthe kernel of the action of NG(T ) on M , and if the representation isfaithful, then they coincide.
(ii) Consider an arbitrary finite-dimensional faithful representation for G(which always exists by Theoreom 5.4.6). Then by (i), W (G, T ) actsfaithfully on the finite set M , and is thus isomorphic to a subgroup ofSym|M |. In particular, W (G, T ) is a finite group. �
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Example 9.3.3. Let G = GLn, and consider the k-dimensional torus
T = {diag(a1, . . . , ak, 1, . . . , 1) | a1, . . . , ak ∈ k×}.
ThenNG(T ) = Monk×GLn−k, whereas CG(T ) = Dk×GLn−k. HenceW (G, T ) ∼=Symk.
We end this chapter by mentioning a result that we will need later (appliedin the case when T is a k-torus).
Theorem 9.3.4 (Rigidity of tori). Let k be an arbitrary field, let G be aconnected linear algebraic k-group, and let T be a linear algebraic k-group ofmultiplicative type. Assume that G acts on T by automorphisms. Then thisaction is trivial.
Proof omitted. �
Remark 9.3.5. The rigidity of tori is a generalization of Theorem 9.3.2.Indeed, when we apply it to the situation where T is a subtorus of G, thenNG(T ) acts on T by inner automorphisms. Passing to the identity componentNG(T )◦ then implies that the action of the connected group NG(T )◦ on Tis trivial, i.e. NG(T )◦ ≤ CG(T ). Hence NG(T )◦ = CG(T )◦, and in particularNG(T )/CG(T ) is a finite group.
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Chapter 10 Solvable linear
algebraic groups
We now come to the study of solvable linear algebraic groups. Giving acomplete classification of such groups is beyond hope, but as we will see, wewill nevertheless be able to prove some strong structure theorems for thisclass of algebraic groups.
We will then study solvable subgroups of general linear algebraic groups;this will lead us to the theory of Borel subgroups, which will play an impor-tant role in our later understanding of reductive groups.
10.1 The derived subgroup of a linear alge-
braic group
To give a rigorous definition of solvable linear algebraic groups, we need thenotion of a derived subgroup, which is similar to but more delicate than thedefinition for concrete groups.
Definition 10.1.1. Let G be a linear algebraic k-group. Then we define thederived subgroup D(G) of G as the intersection of all closed normal subgroupsN of G for which G/N is abelian.
Remark 10.1.2. Recall from section 5.3 that quotients of linear algebraicgroups are a delicate matter. Fortunately, if G is a smooth linear algebraicgroup over an algebraically closed field k, then (G/N)(k) ∼= G(k)/N(k) foreach closed normal subgroup N of G.
We will now give an explicit construction of the derived subgroup of anylinear algebraic k-group. Recall that if Γ is a concrete group, then the derivedsubgroup D(Γ) is
D(Γ) = 〈[g, h] | g, h ∈ Γ〉,
where [g, h] = ghg−1h−1 is the commutator1 of g and h.
1Note that [g, h] is sometimes defined to be g−1h−1gh, but the definition we have chosenagrees with the fact that our group actions are written as left actions.
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Construction 10.1.3. LetG be a linear algebraic k-group, and letA = k[G].For each n ∈ Z≥0, we define the map (between k-functors)
ψn : G2n → G : (g1, h1, . . . , gn, hn) 7→n∏i=1
[gi, hi]
for all gi, hi ∈ GR, for all R ∈ k-alg. There are corresponding maps
ψ∗n : A→ A⊗2n.
Let In := ker(ψ∗n); then we have a descending chain of ideals
I1 ⊇ I2 ⊇ · · · ⊇ In ⊇ . . .
Let I := ∩n≥1In; then I is again an ideal in A. We claim that I is, in fact, thedefining ideal of some closed subgroup of G, i.e. a Hopf ideal of A. (Noticethat the ideals In are not Hopf ideals in general, since the maps ψn are nomorphisms.)
In order to prove our claim, notice that the product of two elementsin im(ψn) is contained in im(ψ2n), and hence for every f ∈ I2n, we have∆(f) ∈ In⊗In. Hence the comultiplication ∆ on A induces a homomorphism
A/I2n∆−−→ A/In ⊗ A/In
for all n, and therefore a homomorphism
A/I∆−−→ A/I ⊗ A/I,
making A/I into a Hopf algebra. Since I is the kernel of the canonical Hopfalgebra epimorphism A→ A/I, this proves that I is a Hopf ideal.
The closed subgroup corresponding to the Hopf ideal I is precisely thederived subgroup D(G).
Remark 10.1.4. In a similar way, we can construct the commutator [H1, H2]for all closed subgroups H1, H2 of a linear algebraic k-group G.
Proposition 10.1.5. Let G be a smooth connected linear algebraic groupover an algebraically closed field k. Then D(G) is also smooth and connected.
Proof. Let A = k[G], and let In and I be as in Construction 10.1.3. No-tice that G is smooth if and only if A is reduced, i.e. has no non-trivialnilpotents, and that G is connected if and only if A has no non-trivialidempotents. Now observe that the map ψ∗n induces an injective mappingA/In → A⊗2n ∼= k[G2n]. Since G2n is smooth and connected, k[G2n] has nonon-trivial nilpotents nor idempotents, and hence the same holds for A/In,for all n. We conclude that A/I has no non-trivial nilpotents or idempotentseither, and hence D(G) is smooth and connected. �
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We now come to the definition of nilpotent and solvable linear algebraicgroups.
Definition 10.1.6. Let G be a linear algebraic k-group.
(i) Let D0(G) := G and Di(G) := D(Di−1(G)) inductively for all i ≥ 1.Then
G = D0(G) ≥ D1(G) ≥ D2(G) ≥ . . .
is called the derived series of G.
(ii) Let D[0](G) := G and D[i](G) := [G,D[i−1](G)] inductively for all i ≥ 1.Then
G = D[0](G) ≥ D[1](G) ≥ D[2](G) ≥ . . .
is called the lower central series series of G.
(iii) The k-group G is called solvable if Dn(G) = 1 for some n.
(iv) The k-group G is called nilpotent if D[n](G) = 1 for some n.
Remark 10.1.7. If k is algebraically closed, then D(G)(k) ∼= D(G(k)), butthis is false in general. For instance, if k = F2 and G = SL2, then D(G) = G,and hence D(G)(k) ∼= SL2(F2), but
D(G(k)) = D(SL2(F2)) ∼= D(Sym3) 6∼= Sym3 .
10.2 The structure of solvable linear algebraic
groups
In this section, we will always assume that G is a smooth linear algebraicgroup over an algebraically closed field k. A crucial fact about connectedsolvable groups (that we will not prove here) is the so-called Borel fixpointtheorem; a consequence of this result is the Lie–Kolchin theorem, statingthat such a group can always be represented by upper-triangular matrices.We will only indicate this approach, and we will instead give a self-containedproof below.
In order to state the Borel fixpoint theorem, we first have to introducethe notion of a complete variety.
Definition 10.2.1. An algebraic variety2 Z is complete if for every variety Y ,
2We have not given a formal definition of an algebraic variety, and we will not attemptto do so (avoiding phrases such as “integral, separated scheme of finite type over analgebraically closed field”). It will be sufficient for our purposes to understand what aprojective variety is: it is the set of zeroes V (S) in projective space Pn for a set S ofhomogeneous polynomials in n+ 1 variables.
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the projection map π : Y ×Z → Y is a closed map, i.e. it maps closed subsetsto closed subsets.
Examples 10.2.2. (1) The affine line A1 is not complete. For instance, theclosed subset S = {(x, y) ∈ A2 | xy = 1} of A2 = A1 × A1 is projectedonto the non-closed subset π(S) = {x ∈ A1 | x 6= 0} of A1.
(2) It turns out that for each integer n ≥ 1, the projective space Pn is acomplete variety. Since a closed subvariety of a complete variety is againcomplete, this implies that in fact every projective variety is complete.
We will mention a couple of useful facts about complete varieties, someof which we will need later.
Proposition 10.2.3. (i) A closed subvariety of a complete variety is com-plete.
(ii) Every projective variety is complete.
(iii) If X is complete, then its image under any morphism X → Y is closedand complete.
(iv) An affine variety is complete if and only if it has dimension zero, i.e.if and only if it is a finite set of points.
Proof omitted. �
We can now state the important Borel fixed-point theorem.
Theorem 10.2.4 (Borel fixed-point theorem). Let G be a smooth connectedsolvable linear algebraic group over an algebraically closed field k, acting ona non-empty complete k-variety X. Then this action has a fixed point, i.e.there is an x ∈ X fixed by G.
Proof omitted. �
Theorem 10.2.5 (Lie–Kolchin theorem). Let G be a smooth connected solv-able linear algebraic group over an algebraically closed field k, and let V bea finite-dimensional G-representation. Then V has a basis such that G isupper-triangular.
Proof. Using Borel’s fixed-point theorem, there are two different ways toprove this. The first approach is inductive. Observe that through the G-rep-resentation, G acts on the projective space Pn−1, where n = dimk V . Sinceprojective space is a complete variety, Borel’s fixed-point theorem implies
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that there is a fixed point x ∈ Pn−1 for the G-action, i.e. there is a one-dimensional subspace of V which is stabilized by G. Take the first basisvector of V to be a generator of this subspace, and proceed by induction.
The second approach is direct, and quite elegant, but requires some fa-miliarity with Grassmann varieties. Let F(V ) be the flag variety of V , i.e.the variety with as points the maximal flags V1 ⊂ V2 ⊂ · · · ⊂ Vn (withdimk Vi = i for each i), viewed as a subvariety of the projective varietyGr1(V ) × · · · × Grn(V ), where Gri(V ) is the Grassmann variety consistingof the i-dimensional subspaces of V . Then F(V ) is a complete variety, andhence the induced action of G on F(V ) has a fixed point, i.e. G(Vi) = Vi forall i. With respect to the corresponding basis of V , the group G is upper-triangular. �
Historically, the Borel fixed-point theorem was a generalization of the Lie–Kolchin triangularization theorem, so in a sense we have been cheating toprove Lie–Kolchin’s theorem using Borel’s theorem for which we omitted theproof. It is instructive to look at a direct proof of the Lie–Kolchin theorem.We first need a lemma.
Lemma 10.2.6. Let V be a vector space over an algebraically closed field k,and let S be a set of commuting elements in Endk(V ). Then there exists abasis for V such that all elements of S are upper-triangular.
Proof. We leave the proof of this fact as an exercise. Use induction on dimk V ,and use the fact that if some f ∈ S is not a scalar multiple of the identity,then f has an eigenspace U 6= V . Show that U is stable under all elementsof S, and apply the induction hypothesis on U and V/U . �
Direct proof of Theorem 10.2.5. As we pointed out above, it suffices to showthat the elements of G(k) have a common eigenvector, because then we canapply induction on the dimension of V . We will prove this by induction onthe length of the derived series of G. If G is commutative, then the resultfollows from Lemma 10.2.6. (Notice that Lemma 10.2.6 shows that there isa basis for V such that G(k) ≤ Tn(k), but since G is smooth, this impliesthat G ≤ Tn.)
Assume now that G is not commutative, and apply the induction hy-pothesis on D(G) to deduce that the elements of N := D(G) have a commoneigenvector. This means that there is some character χ of N for which thespace
Vχ = {v ∈ V | g.v = χ(g)v for all g ∈ N}is non-trivial. Let S be the non-empty set of all χ ∈ X(N) for which Vχ 6= 0,and let W be the sum of all eigenspaces Vχ for χ ∈ S. Then W has a finite
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direct sum decomposition
W =⊕χ∈S
Vχ.
Since G normalizes N , the same computation as in (9.4) shows that G(k)permutes the subspaces Vχ, χ ∈ S.
Now choose χ ∈ S arbitrarily, and let H ≤ G(k) be the stabilizer of Vχ.Since S is finite, H is a finite index subgroup of G(k). We claim that, infact, H = G(k). Notice that
H = {g ∈ G(k) | χ(n) = χ(g−1ng) for all n ∈ N(k)},
which is an algebraic condition3, i.e. H is a closed subgroup of G(k). SinceG(k) is connected and H has finite index, this implies that H = G(k) asclaimed. It follows that G(k) stabilizes Vχ. In particular, there is a represen-tation ρ : G→ GL(Vχ) ∼= GLd, where d = dimVχ.
Next, we claim that N(k) acts trivially on Vχ. For each n ∈ N(k) andeach v ∈ Vχ, we have ρ(n).v = χ(n)v, with χ(n) ∈ k. Since n is a productof commutators of elements of G(k), and since every commutator in GLdhas determinant 1, this implies that ρ(n) has determinant 1, and thereforeχ(n)d = 1. We deduce that the image of the character χ : N → Gm takesvalues in µd ≤ Gm. If char(k) = 0 or char(k) = p - d, then µd is etale, andsince N is connected, it follows from Proposition 8.4.7(ii) that χ is trivial.If p | d, then this argument shows that the image of χ is contained in µpr(where pr is the highest p-power dividing d), but since µpr(k) = 1, we canagain conclude that the action of N(k) on Vχ is trivial. This proves the claimthat N(k) acts trivially on Vχ in all cases.
Therefore, there is an induced action ofG(k)/N(k) on Vχ. SinceG(k)/N(k)is an abelian group, we know that its elements have a common eigenvectorin Vχ, and this is then also a common eigenvector for the elements of G(k),which is what we had to prove. �
Remark 10.2.7. Each of the four hypotheses “smooth”, “connected”, “solv-able” and “algebraically closed” is needed; the theorem becomes false as soonas one of these hypotheses is omitted.
As a consequence of the Lie–Kolchin theorem, we can prove that theset of unipotent elements in a connected solvable group behaves nicely. Forcommutative groups, we had already encountered this in Theorem 6.2.7.
3Notice that χ is a morphism of algebraic groups, so expressing that χ(n) = χ(g−1ng)for a specific n ∈ N(k) is an algebraic condition, i.e. it is a polynomial condition w.r.t. agiven embedding in affine space. The intersection of (infinitely many) algebraic varietiesis again an algebraic variety, so H is algebraic.
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Corollary 10.2.8. Let G be a smooth connected solvable linear algebraicgroup over an algebraically closed field k. Then the set Gu is a closed con-nected nilpotent normal subgroup, and the quotient G/Gu is a torus.
Proof. Without loss of generality, we may assume by Theorem 10.2.5 thatG is a closed subgroup of Tn ≤ GLn. Then g ∈ G(k) is a unipotent elementif and only if it is a unipotent matrix in GLn(k), i.e. if and only if all itsdiagonal elements are equal to 1. Hence the map
ϕ : G→ Gnm : g 7→ diag(g)
is a morphism, with kernel Gu, and hence Gu is a closed normal subgroupof G. Now observe that Gu is a subgroup of Un ≤ GLn, which is easily seento be nilpotent; hence Gu is nilpotent itself. Next, notice that T = G/Gu isisomorphic to some subgroup of Gn
m, and since it is smooth and connected(as a quotient of a smooth connected group), it follows that it is a torus.
We finally show that Gu is connected. Observe that when G is abelian,Gu is a quotient of G, which is therefore connected (see Theorem 6.2.7). Thisimplies that (G/D(G))u is connected. On the other hand, D(G) ≤ Gu, andthe exact sequence
1→ Gu/D(G)→ G/D(G)→ T → 1
shows that every unipotent element of G/D(G) is contained in Gu/D(G),and hence Gu/D(G) = (G/D(G))u is connected. Since D(G) is itself alsoconnected (see Proposition 10.1.5), it follows by Corollary 8.4.8 that Gu isconnected as well. �
With some more effort, one can show the following structure theorem forsolvable groups.
Theorem 10.2.9. Let G be a smooth connected solvable linear algebraicgroup over an algebraically closed field k. Then:
(i) Gu is a closed connected nilpotent normal subgroup.
(ii) The quotient G/Gu is a torus.
(iii) There is a series of closed subnormal subgroups
1 = N0 ≤ N1 ≤ · · · ≤ Nd = Gu
of G, such that for each i ∈ {1, . . . , d}, the group Ni−1 is normal in Ni
and Ni/Ni−1∼= Ga.
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(iv) The extension1→ Gu → G→ G/Gu → 1
is split, i.e. Gu has a complement in G. Moreover, any two such com-plements are conjugate in G.
(v) Every semisimple element s ∈ G(k) is contained in a complement ofGu in G. Moreover, CG(s) is smooth and connected.
Proof. We have already shown (i) and (ii), and we omit the proof of the otherfacts. �
Definition 10.2.10. Let G be a smooth connected solvable linear algebraicgroup over an algebraically closed field k. Every complement of Gu is called amaximal torus of G. Notice that such a complement is indeed a torus since itis isomorphic to G/Gu, and it is maximal w.r.t. inclusion since any subgroupof G properly containing it, would intersect Gu non-trivially and hence cannot be a torus.
The following corollary shows in particular that every torus is containedin a maximal torus.
Corollary 10.2.11. Let G be a smooth connected solvable linear algebraicgroup over an algebraically closed field k, and let H be a commutative sub-group of G consisting of semisimple elements only. Then H is contained ina maximal torus. Moreover, any two such maximal tori are conjugate by anelement of CG(H).
Proof. We will prove the result by induction on dim(G). Notice that the re-sult is obvious if all elements of H are central in G, because a central semisim-ple element is contained in every maximal torus by Theorem 10.2.9(iv) and (v),and CG(H) = G in this case.
So assume that there is some h ∈ H \ Z(G). Then CG(h) is a propersmooth connected subgroup of G, and H ≤ CG(h). Since G is smooth andconnected, dimCG(h) < dimG. By the induction hypothesis, H is containedin a maximal torus S of CG(h), and any two such maximal tori S and S ′ areconjugate by an element of CCG(h)(H) = CG(H).
It remains to show that S (and then also S ′) is a maximal torus in G.By Theorem 10.2.9(v), h is contained in a maximal torus T of G, so inparticular T ≤ CG(h), and hence T is also a maximal torus of CG(h). ByTheorem 10.2.9(iv) applied on CG(h), S and T are conjugate in CG(h), andhence S is also a maximal torus of G. �
We mention the following classification result, which apart from the the-ory we have just seen, also requires a good deal of algebraic geometry.
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Theorem 10.2.12. Let G be a smooth connected 1-dimensional linear alge-braic group over an algebraically closed field k. Then G ∼= Ga or G ∼= Gm.
Proof omitted. �
10.3 Borel subgroups
We now go back to the situation where G is a general (smooth) linear alge-braic group over an algebraically closed field k. As we will see, understandingthe solvable subgroups inside G will be important for determining the struc-ture of G. This brings us to the notion of Borel subgroups.
Definition 10.3.1. Let G be a smooth linear algebraic group over an alge-braically closed field k. A Borel subgroup of G is a maximal closed smoothconnected solvable subgroup of G.
The following fact is an important feature of algebraic groups, but itsproof requires more theory than we have covered.
Theorem 10.3.2. Let G be a smooth linear algebraic group over an alge-braically closed field k, and let B be a Borel subgroup of G. Then G/B is aprojective k-variety.
Proof omitted. �
Together with Borel’s fixed-point theorem (Theorem 10.2.4), it has thefollowing corollary.
Corollary 10.3.3. Let G be a smooth linear algebraic group over an alge-braically closed field k, and let B and B′ be two Borel subgroups of G. ThenB and B′ are conjugate.
Proof. Consider the action of B on the projective variety G/B′ which isgiven by left multiplication on the set of left cosets of B′ in G. Then byBorel’s fixed-point theorem, this action has a fixed point, i.e. there is someleft coset gB′ which is stable under left multiplication by B, i.e. bgB′ = gB′
for all b ∈ B. This implies g−1Bg ⊆ B′. Since both g−1Bg and B′ are Borelsubgroups of G, the maximality of both implies that g−1Bg = B′. �
We have seen that maximal tori inside solvable groups play an importantrole, but this is also true for general linear algebraic groups.
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Definition 10.3.4. Let G be a smooth connected linear algebraic group overan algebraically closed field k. A maximal torus of G is a torus of G whichis maximal (w.r.t. inclusion).
Proposition 10.3.5. Let G be a smooth connected linear algebraic groupover an algebraically closed field k. Then all maximal tori in G are conjugate.Even more, G acts transitively by conjugation on the set{
(T,B) | B a Borel subgroup of G, T a maximal torus in B}.
Proof. Notice that a torus is a closed connected solvable subgroup, so bydefinition, every torus T ofG is contained in some Borel subgroup B ofG, andif T is a maximal torus in G, then it is also a maximal torus in B. The resultnow follows from Corollary 10.3.3 together with Theorem 10.2.9(iv). �
Remark 10.3.6. When k is not algebraically closed, it is no longer true thatall maximal tori of G are conjugate, but by Proposition 10.3.5 they becomeconjugate after base change to k. The classification of maximal tori over kthus becomes an “arithmetic problem” determined by k/k, which is typicallystudied using Galois cohomology.
Since all maximal tori are conjugate, the dimension of a maximal torusis a well-defined number.
Definition 10.3.7. Let G be a smooth connected linear algebraic group overan algebraically closed field k. Then the rank rk(G) of G is defined to be thedimension of a maximal torus in G.
We will now further illustrate the importance of Borel subgroups by in-dicating that their structure already determines some of the structure of G.For instance, the center of B essentially determines the center of G, and ifB is nilpotent, then already G was nilpotent to begin with. We start with alemma.
Lemma 10.3.8. Let G be a smooth linear algebraic group over an alge-braically closed field k, and let B be a Borel subgroup of G.
(i) If G is connected and G 6= 1, then B 6= 1.
(ii) The index [NG(B) : B] is finite.
(iii) NG(NG(B)) = NG(B).
Proof. (i) Assume B = 1. Then G = G/B is simultaneously an affinevariety and a projective variety. Since G is connected, this can only betrue if G = 1.
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(ii) Let H = NG(B)◦ be the connected component of the normalizer of Bin G. Then B is a Borel subgroup of H, which is normal. This impliesthat H/B is simultaneously an affine variety and a projective variety,which can only be true if H/B = 1, and hence B = H = NG(B)◦. Itfollows that B is a finite index subgroup of NG(B).
(iii) Assume that g ∈ G normalizes NG(B). Then it also normalizes theidentity component NG(B)◦ = B. �
Proposition 10.3.9. Let G be a smooth connected linear algebraic groupover an algebraically closed field k, and let B be a Borel subgroup of G. Then
Z(G)◦ ≤ Z(B) ≤ Z(G).
Proof. Notice that Z(G)◦ is a closed connected solvable subgroup, so bydefinition, it is contained in some Borel subgroup B′ of G. Since B′ is con-jugate to B and conjugation acts trivially on Z(G)◦, this implies that in factZ(G)◦ ≤ B, and consequently Z(G)◦ ≤ Z(B).
Now let z ∈ Z(B) be arbitrary. Consider the map
ϕ : G→ G : g 7→ gzg−1.
Then ϕ is constant on every left B-coset, i.e. ϕ(gb) = ϕ(gb′) for all g ∈ Gand all b, b′ ∈ B. Therefore, ϕ induces a morphism (of algebraic varieties)
ϕ : G/B → G : gB 7→ gzg−1.
Because G/B is a projective variety and G is an affine variety, Proposi-tion 10.2.3 implies that the map ϕ has finite image, and hence ϕ has finiteimage as well. Since G and hence also G/B is connected, this implies that ϕis a constant map. Hence z ∈ Z(G). �
Proposition 10.3.10. Let G be a smooth connected linear algebraic groupover an algebraically closed field k, and let B be a Borel subgroup of G. If Bis nilpotent, then G is nilpotent.
Proof. We prove the result by induction on dim(G). It is trivial whendim(G) = 0, so assume dim(G) ≥ 1, which implies by Lemma 10.3.8(i)that dim(B) ≥ 1 as well. Since B is nilpotent, the last non-trivial term ofthe lower central series of B is a connected non-trivial central subgroup Nof B, and hence dim(Z(B)) ≥ 1. By Proposition 10.3.9, this implies thatdim(Z(G)) ≥ 1 as well, and hence the dimension of G/Z(G)◦ is strictly lowerthan dim(G). Notice that Z(G)◦ ≤ B, hence B/Z(G)◦ is a Borel subgroupof G/Z(G)◦, which is nilpotent. By induction, G/Z(G)◦ is nilpotent, andhence G is nilpotent as well. �
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Chapter 11 Semisimple and
reductive groups
We continue with our assumption that G is a smooth linear algebraic groupover an algebraically closed field k. Our aim is to understand the structure ofsuch a G in its full generality, but as we have seen, already our understandingof unipotent, or more generally solvable, linear algebraic groups, is limited,in the sense that there is no hope of classifying such groups.
On the other hand, we will see that when we get rid of the unipotent orsolvable “part” of a linear algebraic group G, then we are left with a so-calledreductive or semisimple group, respectively, and it will turn out that we havea very good understanding of such groups: we will be able to classify them.An essential ingredient of the structure of such a group will be given by itsso-called root datum.
11.1 Semisimple and reductive linear algebraic
groups
We first introduce the notions of reductive and semisimple groups. We beginby stating the following useful fact.
Lemma 11.1.1. Let G be a linear algebraic group over an algebraicallyclosed field k. Let H be a closed subgroup of G and let N be a closed nor-mal subgroup. If H and N are solvable (resp. unipotent, resp. connected,resp. smooth), then HN is solvable (resp. unipotent, resp. connected, resp.smooth).
Proof. In each case, use the fact that HN/N ∼= H/(H ∩ N), and that HNis an extension of HN/N by N , together with the fact that these properties(solvable, unipotent, connected, smooth) are preserved by quotients and byextensions. In each case, this requires a different argument, and we omit thedetails. �
Definition 11.1.2. Let G be a smooth linear algebraic group over an alge-braically closed field k.
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(i) The radical R(G) is the largest smooth closed connected solvable normalsubgroup of G.
(ii) The unipotent radical Ru(G) is the largest smooth closed connectedunipotent normal subgroup of G; it coincides with R(G)u, the unipotentpart of R(G).
(iii) We call G semisimple if R(G) is trivial.
(iv) We call G reductive if Ru(G) is trivial.
The following characterizations are useful.
Lemma 11.1.3. Let G be a smooth linear algebraic group over an alge-braically closed field k.
(i) G is semisimple if and only if G does not have non-trivial smooth closedconnected commutative normal subgroups.
(ii) G is reductive if and only if R(G) is a torus.
(iii) G is reductive if and only if the only non-trivial smooth closed connectedcommutative normal subgroups of G are tori.
Proof. (i) Assume that G does not have non-trivial smooth closed con-nected commutative normal subgroups, and suppose that G is notsemisimple, i.e. R(G) 6= 1. Notice that both R(G) and D(G) are char-acteristic subgroups of G, so each group occuring in the derived seriesof R(G) is itself a smooth closed connected normal subgroup of G. Thelast non-trivial term of this series is commutative, which contradictsour assumption.
(ii) Notice that Ru(G) = R(G)u, so G is reductive if and only if R(G) isa smooth connected solvable group with trivial unipotent part. By thestructure theorem of solvable groups (Theorem 10.2.9), this is equiva-lent to the fact that R(G) is a torus.
(iii) Assume that every non-trivial smooth closed connected commutativenormal subgroup of G is a torus. Suppose that G is not reductive, i.e.Ru(G) 6= 1. As in the proof of (i), each group occuring in the derivedseries of Ru(G) is itself a smooth closed connected normal subgroupof G. The last non-trivial term of this series is commutative, and byour assumption, it is a torus. This contradicts the fact that Ru(G) is anon-trivial unipotent group. �
The name “semisimple” might sound mysterious at this point. Our nextaim is to explain that semisimple groups are, in some sense, closely relatedto simple linear algebraic groups.
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Definition 11.1.4. Let G be a linear algebraic k-group.
(i) We call G simple if it is smooth, connected, non-commutative, and hasno non-trivial proper normal subgroups.
(ii) We call G almost-simple if it is smooth, connected, non-commutative,and has no infinite proper normal subgroups.
(iii) We say that G is the almost-direct product of its closed subgroupsG1, . . . , Gr if the product map
G1 × · · · ×Gr → G : (g1, . . . , gr) 7→ g1 · · · gr
is a surjective homomorphism with finite kernel. (In particular, thesubgroups Gi are normal in G, and they pairwise commute.)
Clearly, an almost-direct product of almost-simple linear algebraic groupsis semisimple. The converse is also true:
Theorem 11.1.5. Let G be a semisimple linear algebraic group over analgebraically closed field k. Then G is an almost-direct product of its almost-simple closed subgroups, namely the minimal closed connected infinite normalsubgroups of G. (These are called the almost-simple factors of G.)
Proof omitted. �
Corollary 11.1.6. Let G be a semisimple linear algebraic group over analgebraically closed field k. Then:
(i) Every quotient of G is semisimple.
(ii) If N is a smooth connected normal subgroup of G, then N is the productof the almost-simple factors contained in N , and is centralized by theremaining ones. In particular, N is semisimple.
(iii) G is perfect, i.e. D(G) = G.
(iv) The center of G is a finite group of multiplicative type.
Proof. Statements (i) and (ii) are immediate from Theorem 11.1.5. To prove(iii), it suffices to observe that D(H) = H for any almost-simple group H,which is obvious from the definitions. (Recall that D(H) is smooth andconnected, and hence cannot be a non-trivial finite group.) Finally, to prove(iv), observe that1 Z(G)◦red is a closed smooth connected commutative normal
1When G is an algebraic group over an algebraically closed field k, which is not neces-sarily smooth, then we can smoothen it as in Remark 8.5.16 to obtain a closed subgroupGred of G. (In general, Gred need not be normal in G!)
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subgroup of G, and hence Z(G)◦red ≤ R(G) = 1, which shows that Z(G) is afinite group. �
We now show that reductive groups are, in a precise sense, not too faraway from semisimple groups.
Theorem 11.1.7. Let G be a connected reductive linear algebraic group overan algebraically closed field k. Then
R(G) = Z(G)◦red and G = R(G)D(G).
Moreover, R(G) ∩ D(G) is finite, and D(G) is semisimple.
Proof. We will first show that R(G) = Z(G)◦red. Since Z(G)◦red is a closedsmooth connected commutative normal subgroup of G, it is certainly con-tained in R(G). Conversely, the fact that G is reductive implies that R(G)is a torus. Notice that G acts on R(G) by conjugation. By rigidity oftori (see Theorem 9.3.4), this action is trivial, i.e. R(G) ≤ Z(G). SinceR(G) is smooth and connected, this implies R(G) ≤ Z(G)◦red and henceR(G) = Z(G)◦red.
Next, notice that R(G)D(G) is a normal subgroup of G. Then the groupG/R(G)D(G) is a quotient of the commutative group G/D(G), and a quo-tient of the semisimple group G/R(G); hence G/R(G)D(G) is a commutativesemisimple group, which is therefore trivial. It follows that G = R(G)D(G).
Our next step is to show that R(G) ∩ D(G) is finite. Write T = R(G) =Z(G)◦red, and notice that T is a diagonalizable subgroup of G. Consider afinite-dimensional faithful representation G ↪→ GLV , and use Theorem 9.2.8to write
V =⊕
χ∈X(T )
Vχ.
Since T is central in G, the elements of G stabilize each Vχ; hence there isan induced monomorphism
α : G→ GL(Vχ1)× · · · × GL(Vχr),
where χ1, . . . , χr are the characters of T for which Vχ 6= 0. Hence
α(D(G)) ≤ SL(Vχ1)× · · · × SL(Vχr).
On the other hand, by definition of the eigenspaces Vχ,
α(T ) ≤ Sc(Vχ1)× · · · × Sc(Vχr),
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where Sc(Vχ) denotes the group of scalar matrices of GL(Vχ). Since SL(Vχ)∩Sc(Vχ) is finite for each χ, it follows that α(T ∩ D(G)) is finite, and since αis a monomorphism, this implies that T ∩ D(G) is also finite.
We finally show that D(G) is semisimple. Notice that the homomorphismD(G) → G/R(G) is surjective, and has finite kernel R(G) ∩ D(G). SinceG/R(G) is semisimple, this implies that D(G) is semisimple as well. �
Remark 11.1.8. In what follows, we will be mainly interested in reductivegroups, and not just in semisimple groups. One of the reasons is that manynaturally occuring groups, such as GLn, are reductive but not semisimple.Another reason is that the centralizer of a torus in a semisimple group isusually not semisimple, but it is reductive again, and these centralizers areimportant subgroups for various reasons. See, for example, Proposition 11.2.8below.
Recall that the rank of a smooth linear algebraic group over an alge-braically closed field was defined to be the dimension of a maximal torus.For reductive groups, the following related notion is sometimes more natu-ral.
Definition 11.1.9. Let G be a connected reductive linear algebraic groupover an algebraically closed field k. Then the semisimple rank of G is definedto be the rank of its derived group D(G) (which is a semisimple group byTheorem 11.1.7).
Example 11.1.10. The rank of G = GLn is equal to n, but its semisimplerank is equal to the rank of D(G) = SLn, which is n− 1.
11.2 The root datum of a reductive group
To each reductive linear algebraic group (over an algebraically closed field k),we will attach a combinatorial object, called the root datum, which will de-termine G uniquely up to isomorphism. (More precisely, it will be associatedto a pair (G, T ), where G is a reductive group, and T is a maximal torusof G.) Before we introduce this combinatorial object, we recall the notionof characters, we introduce cocharacters, and we define a pairing betweencharacters and cocharacters.
Definition 11.2.1. Let T be a torus over an algebraically closed field k.
(i) A character of T is a morphism χ : T → Gm.
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(ii) The character group of T is the free abelian group
X(T ) := {characters of T},
where the group addition is given by (χ1 + χ2)(g) = χ1(g)χ2(g) for allg ∈ T .
(iii) A cocharacter of T is a morphism λ : Gm → T .
(iv) The cocharacter group of T is the free abelian group
Y (T ) := {cocharacters of T},
where the group addition is given by (λ1 + λ2)(g) = λ1(g)λ2(g) for allg ∈ T .
(v) We define a pairing
〈·, ·〉 : X(T )× Y (T )→ End(Gm) ∼= Z : (χ, λ) 7→ 〈χ, λ〉 := χ ◦ λ.
Hence for all t ∈ Gm(R) = R×, we have χ(λ(t)) = t〈χ,λ〉.
Example 11.2.2. Let T = Dn be the group of invertible diagonal n-by-n ma-trices. Then X(T ) is a free abelian group of rank n, with basis (χ1, . . . , χn),where
χi : Dn → Gm : diag(a1, . . . , an) 7→ ai.
Similarly, Y (T ) is a free abelian group of rank n, with basis (λ1, . . . , λn),where
λi : Gm → Dn : t 7→ diag(1, . . . , t, . . . , 1)
(where the t is on the i-th position). The pairing between X(T ) and Y (T )is then given by
〈χi, λj〉 = δij,
where δij is the Kronecker delta.
We are now ready to introduce the notion of roots in a reductive linearalgebraic group. In order to define it, we recall that a linear algebraic groupG acts on its Lie algebra g = Lie(G) through its adjoint representation
Ad: G→ GLg.
Definition 11.2.3. Let G be a reductive linear algebraic group over an alge-braically closed field k, and let T be a maximal torus in G. By Theorem 9.2.8,the Lie algebra g has a decomposition
g = g0 ⊕⊕
χ∈X(T )\{0}
gχ,
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wheregχ = {v ∈ g | g.v = χ(g)v for all g ∈ T}
for all χ ∈ X(T ); in particular,
g0 = {v ∈ g | g.v = v for all g ∈ T}
is the subspace of elements of g fixed by T .
A root of (G, T ) is defined to be a non-zero character χ ∈ X(T ) \ {0} forwhich gχ 6= 0. Notice that the set of roots is a finite subset of X(T ), whichwe will denote by R(G, T ).
We will illustrate this concept with several examples. Notice that theset R(G, T ) is a subset of X(T ), which is a free abelian group of finite rank(say n); hence we can view it as a subset of the Euclidean space Rn throughthe isomorphism X(T ) ∼= Zn ⊂ Rn, which will allow us to visualize the setof roots.
Examples 11.2.4. (1) Let G = GL2. Then g = gl2 = Mat2(k), with[A,B] = AB −BA. Consider the maximal torus
T =
{(x1
x2
)| x1x2 6= 0
}.
Then X(T ) = Zχ1 ⊕ Zχ2, where
(aχ1 + bχ2).
(x1
x2
)= xa1x
b2
for all a, b ∈ Z. By the definition of the adjoint representation, T actson g by conjugation:(
x1
x2
)(a bc d
)(x−1
1
x−12
)=
(a x1
x2b
x2x1c d
).
This shows that g has a decomposition
g = g0 ⊕ gχ1−χ2 ⊕ gχ2−χ1 ,
where dim g0 = 2 and dim gχ1−χ2 = dim gχ2−χ1 = 1. Hence
R(G, T ) = {α,−α} where α = χ1 − χ2.
When we identify X(T ) with Z2 ⊂ R2, we get
R(G, T ) = {±(e1 − e2)} ⊂ Z2 ⊂ R2.
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(2) Let G = SL2. Then g = sl2 = {A ∈ Mat2(k) | tr(A) = 0}. Consider themaximal torus
T =
{(x
x−1
)| x 6= 0
}.
Then X(T ) = Zχ, where
(aχ).
(x
x−1
)= xa
for all a ∈ Z. Again, T acts on g by conjugation:(x
x−1
)(a bc −a
)(x−1
x
)=
(a x2b
x−2c −a
).
This shows that g has a decomposition
g = g0 ⊕ g2χ ⊕ g−2χ,
where dim g0 = 1 and dim g2χ = dim g−2χ = 1. Hence
R(G, T ) = {α,−α} where α = 2χ.
When we identify X(T ) with Z1 ⊂ R1, we get
R(G, T ) = {±2e1} ⊂ Z1 ⊂ R1.
(3) Let G = PGL2 = GL2/Gm. Then g = gl2/sc2. Consider the maximaltorus
T =
{(x1
x2
)| x1x2 6= 0
}/ Gm.
Then X(T ) = Zχ, where
(aχ).
(x1
x2
)=
(x1
x2
)a124
for all a ∈ Z. We compute the action of T on g by conjugation:(x1
x2
)(a bc d
)(x−1
1
x−12
)=
(a x1
x2b
x2x1c d
).
This shows that g has a decomposition
g = g0 ⊕ gχ ⊕ g−χ,
where dim g0 = 1 and dim gχ = dim g−χ = 1. Hence
R(G, T ) = {α,−α} where α = χ.
When we identify X(T ) with Z1 ⊂ R1, we get
R(G, T ) = {±e1} ⊂ Z1 ⊂ R1.
(4) Let G = GLn. Then g = gln = Matn(k). Consider the maximal torus
T =
x1
. . .
xn
| x1 · · ·xn 6= 0
.
Then X(T ) = Zχ1 ⊕ · · · ⊕ Zχn, where
(a1χ1 + · · ·+ anχn).
x1
. . .
xn
= xa11 · · ·xann
for all a1, . . . , an ∈ Z. We compute the action of T on g by conjugation:x1
. . .
xn
a11 · · · a1n
.... . .
...an1 · · · ann
x
−11
. . .
x−1n
=
a11 · · · x1xna1n
.... . .
...xnx1an1 · · · ann
(i.e. the matrix with as (i, j)-th entry xi
xjaij). This shows that g has a
decomposition
g = g0 ⊕⊕i 6=j
gχi−χj.
Notice that dim g0 = n and dim gχi−χj= 1 for all i 6= j. Hence
R(G, T ) = {χi − χj | 1 ≤ i, j ≤ n, i 6= j}.
When we identify X(T ) with Zn ⊂ Rn, we get
R(G, T ) = {±(ei − ej) | 1 ≤ i < j ≤ n} ⊂ Zn ⊂ Rn.
For example, for n = 3, we get the following configuration:
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X
Y
Z
Although the set of roots (which we will call a root system) alreadyprovides a lot of information about the group (it will determine the type of thealgebraic group), we will need another piece of data to complete the picture.This is captured by the following definition. Notice that this definition ispurely combinatorial and does not involve any reference to linear algebraicgroups whatsoever.
Definition 11.2.5. A root datum is a quadruple
Ψ = (X,R,X∨, R∨),
where
• X and X∨ are free Z-modules of finite rank, equipped with a bilinearpairing 〈·, ·〉 : X ×X∨ → Z;
• R and R∨ are finite subsets of X and X∨, respectively, equipped witha bijection α↔ α∨, such that:
(i) 〈α, α∨〉 = 2 for all α ∈ R;
(ii) sα(R) ⊆ R for all α ∈ R, where
sα : X → X : x 7→ x− 〈x, α∨〉α;
(iii) the Weyl group
W (Ψ) := 〈sα | α ∈ R〉 ≤ Aut(X)
is a finite group.
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A root datum is called reduced if α ∈ R implies 2α 6∈ R, or equivalently, ifthe only multiples of α ∈ R again contained in R are α and −α.
To get a feeling for these objects, we will show a few properties of themaps sα; they show that, in some sense, the sα are (abstract) reflections.
Proposition 11.2.6. Let Ψ = (X,R,X∨, R∨) be a root datum, and α ∈ R.Then:
(i) sα(α) = −α;
(ii) s2α = idX ;
(iii) sα(x) = x for all x such that 〈x, α∨〉 = 0;
(iv) If q ∈ Q is such that qα ∈ R, then (qα)∨ = 1qα∨. In particular,
(−α)∨ = −α∨, and hence sα = s−α.
Proof. Notice that (i) and (iii) are obvious from the definitions. We will nowshow (ii). So let x ∈ X be arbitrary; then
sα(sα(x)) = sα(x− 〈x, α∨〉α
)= sα(x)− 〈x, α∨〉sα(α)
= x− 〈x, α∨〉α + 〈x, α∨〉α = x.
The last statement (iv) is more involved, and we will omit its proof. �
Our next goal is to attach a root datum to a given reductive linear alge-braic group (together with a given maximal torus). We begin with a lemma.
Lemma 11.2.7. Let G be a reductive linear algebraic group over an alge-braically closed field k, and let T be a maximal torus in G. Then the actionof the group W (G, T ) = NG(T )/CG(T ) on X(T ) stabilizes the set R(G, T )of roots.
Proof. It follows from Theorem 9.3.2 that W (G, T ) acts on the set
M = {χ ∈ X(T ) | gχ 6= 0} = R(G, T ) ∪ {0}.
Since the elements of NG(T ) stabilize the space g0 of fixed vectors, we con-clude that there is an induced action of W (G, T ) on R(G, T ). �
The following result will define the coroots.
Proposition 11.2.8. Let G be a reductive linear algebraic group over analgebraically closed field k, and let T be a maximal torus in G. Let X(T )
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be the character group of T , let Y (T ) be the cocharacter group of T , and letR = R(G, T ) ⊂ X(T ) be the set of roots. For each α ∈ R, we let
Tα := ker(α)◦, Gα := CG(Tα).
Then W (Gα, T ) contains a unique non-trivial element sα, and there is aunique element α∨ ∈ Y (T ) such that
sα(x) = x− 〈x, α∨〉α
for all x ∈ X(T ). Moreover, 〈α, α∨〉 = 2.
Proof. We omit the proof. The crucial point is that Gα is a reductive groupof semisimple rank 1 (see Definition 11.1.9). Since every semisimple groupof rank 1 is isomorphic to either SL2 or PGL2, the other statements can thenbe shown by looking at each of these two cases separately. �
The cocharacter α∨ is called the coroot of α, and the set of all coroots isdenoted by R∨(G, T ).
To each root α, we can associate a so-called root group Uα:
Proposition 11.2.9. Let G be a reductive linear algebraic group over analgebraically closed field k, let T be a maximal torus in G, and let α ∈ R(G, T )be a root. Then:
(i) There is a unique subgroup Uα of G isomorphic to Ga, such that foreach isomorphism uα : Ga → Uα, we have
t · uα(x) · t−1 = uα(α(t)x
),
for all t ∈ T (k) and all x ∈ k = Ga(k).
(ii) The group Uα is the unique subgroup of G normalized by T with Liealgebra gα.
(iii) Let Tα := ker(α)◦ and Gα := CG(Tα) as before. Then Gα = 〈T, Uα, U−α〉.
Proof omitted. �
We have assembled all the necessary facts to prove that we can associatea root datum to any reductive group.
Theorem 11.2.10. Let G be a reductive linear algebraic group over an al-gebraically closed field k, and let T be a maximal torus in G. Then
Ψ(G, T ) :=(X(T ), R(G, T ), Y (T ), R∨(G, T )
)is a reduced root datum.
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Proof. We already know that X(T ) and Y (T ) are free modules of equal rank,equipped with a pairing 〈·, ·〉 (see Definition 11.2.1(v)), and that R = R(G, T )and hence also R∨ = R∨(G, T ) are finite. The fact that 〈α, α∨〉 = 2 holdsfor each root α is contained in Proposition 11.2.8. The same proposition alsoshows that sα ∈ W (Gα, T ) ≤ W (G, T ); since W (G, T ) stabilizes the set R byLemma 11.2.7, this shows that sα(R) ⊆ R. Moreover, the fact that W (G, T )is a finite group (see Theorem 9.3.2) shows that the group 〈sα | α ∈ R〉 isalso finite. (In fact, it coincides with W (G, T ), but it requires more effortto show this.) The fact that the root datum is always reduced, is somewhatmore delicate, and we will omit its proof. �
Before we give examples, we mention the fundamental fact that a reduc-tive linear algebraic group over an algebraically closed field k is completelydetermined by its root datum and the field k only.
Theorem 11.2.11. (i) Let G be a reductive linear algebraic group over analgebraically closed field k, and let T and T ′ be two maximal tori in G.Then Ψ(G, T ) and Ψ(G, T ′) are isomorphic.
(ii) Let k be an algebraically closed field. Each reduced root datum arisesfrom a reductive linear algebraic group over k.
(iii) Let G and G′ be two reductive linear algebraic groups over the samealgebraically closed field k, and let T and T ′ be maximal tori in G andG′, respectively. Assume that Ψ(G, T ) and Ψ(G′, T ′) are isomorphicroot data. Then G and G′ are isomorphic. More precisely, there is anisomorphism from G to G′ mapping T to T ′.
Proof omitted. �
We now come to examples.
Examples 11.2.12. (1) Let G = SL2, and let T be as in Example 11.2.4(2).Then
X = X(T ) = Zχ with χ. ( x x−1 ) = x;
X∨ = Y (T ) = Zλ with λ.t = ( t t−1 ) ;
R = R(G, T ) = {α,−α} with α = 2χ;
R∨ = R∨(G, T ) = {α∨,−α∨} with α∨ = λ.
Observe that (α ◦ α∨)(t) = (2χ ◦ λ)(t) = t2, so indeed 〈α, α∨〉 = 2. Wehave W (G, T ) = W (Ψ) = 〈sα, s−α〉 = 〈sα〉 = {1, sα}. Hence we canwrite
Ψ(SL2, T ) ∼=(Z, {−2, 2},Z, {−1, 1}
),
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with 〈x, y〉 = xy, and where 2∨←→ 1 and −2
∨←→ −1.Notice that ker(α) is a group of order 2; hence Tα = ker(α)◦ = 1, andin particular Gα = G. The unique non-trivial element sα of W (Gα, T ) isthe image of nα := ( 0 1
−1 0 ) ∈ NG(T ) \ CG(T ) in W (G, T ).We claim that the root group Uα is given by
Uα =
{(1 x
1
) ∣∣∣ x ∈ k} .Consider the isomorphism
uα : Ga → Uα : x 7→(
1 x1
).
(This is of course not the only isomorphism from Ga → Uα, but thechoice is irrelevant.) We check that the condition from Proposition 11.2.9is satisfied. So let t = ( s s−1 ) be arbitrary; then indeed
t · uα(x) · t−1 =
(s
s−1
)(1 x
1
)(s−1
s
)=
(1 s2x
1
)= uα
(s2x)
= uα(α(t)x
)for all x ∈ k. Similarly, it can be checked that
U−α =
{(1x 1
) ∣∣∣ x ∈ k} .Notice that G = 〈T, Uα, U−α〉 by Proposition 11.2.9(ii); in fact, G =〈Uα, U−α〉 in this case.
(2) Let G = PGL2, and let T be as in Example 11.2.4(3). In this case, α = χand α∨ = 2λ, and we get
Ψ(PGL2, T ) ∼=(Z, {−1, 1},Z, {−2, 2}
),
with 〈x, y〉 = xy, and where 1∨←→ 2 and −1
∨←→ −2.Observe that the root systems of PGL2 and SL2 are isomorphic, but theirroot data are not.
(3) Let G = Gm; then G is a torus, so we let T = G. Observe that Ghas no roots in this case. (The group G is reductive of rank 1, but hassemisimple rank 0.) We have
Ψ(Gm, T ) ∼=(Z, ∅,Z, ∅
).
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(4) Let G = GLn, and let T be as in Example 11.2.4(4). Define the charactersχi and the cocharacters λi as in Example 11.2.2. Let
R = {αij | i 6= j}, αij := χi − χj;R∨ = {α∨ij | i 6= j}, α∨ij := λi − λj.
Notice that 〈α, α∨〉 = 2 for every α ∈ R. It is not too hard to check that
sαij(χk) =
χj if k = i;
χi if k = j;
χk if k 6= i, j.
Hence sαijacts on the basis {χ1, . . . , χn} of X as the transposition (ij)
on the set {1, . . . , n}. This implies that
W = 〈sα | α ∈ R〉 ∼= Symn .
Observe that the groups Gα = CG(Tα) are isomorphic to GL2 × Gn−2m ;
they have rank n but semisimple rank 1.We now describe the root groups of G (w.r.t. T ). For each i 6= j andeach x ∈ k, we write Eij(x) for the matrix with 1’s on the diagonal, withx on position (i, j), and with 0’s everywhere else. Let
Uij := {Eij(x) | x ∈ k} .
Then it is quickly verified that Uij is the root group corresponding toUαij
, for each i 6= j. Notice that all the root groups Uij with i < jare upper triangular, while all the root groups Uij with i > j are lowertriangular.
11.3 Classification of the root data
In this section, we want to present the classification of root data, mostlywithout proofs. To begin, we distinguish a few degenerate cases.
Definition 11.3.1. Let Ψ = (X,R,X∨, R∨) be a root datum.
(i) We call Ψ a toral root datum if R = R∨ = ∅.(ii) We call Ψ a semisimple root datum if R generates a finite index sub-
group of X.
The reason for this terminology can easily be guessed:
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Proposition 11.3.2. Let G be a reductive linear algebraic group over analgebraically closed field k, let T be a maximal torus in G, and let Ψ =Ψ(G, T ) be the corresponding root datum. Then
(i) Ψ is toral if and only if G is a torus;
(ii) Ψ is semisimple if and only if G is semisimple.
Proof. We omit the proof. The key ingredient is the fact that the center ofG coincides with the intersection
⋂α∈R ker(α). �
From now on, we focus on semisimple root data. The set of roots Ralready contains a lot of structure on its own, although it is not sufficient torecover the algebraic group uniquely up to isomorphism (as we have seen inthe examples SL2 vs. PGL2). This additional structure of the set R is knownas a root system.
Definition 11.3.3. (i) Let V be a finite-dimensional vector space over Q.A subset R of V is called a root system in V , if:
(a) R is finite, spans V (as a vector space), and does not contain 0;
(b) For each α ∈ R, there is a (unique) reflection sα with vector αstabilizing the set R;
(c) For all α, β ∈ R, the element sα(β)− β is an integer multiple of α.
If in addition
(d) For each α ∈ R, the only multiple of α which lies again in R is −α,
then the root system is called reduced.
(ii) The rank of the root system is defined to be the dimension of V .
(iii) If (V1, R1), . . . , (Vn, Rn) are root systems, then the direct sum of theseroot systems is the root system (V1 ⊕ · · · ⊕ Vn, R1 t · · · tRn).
(iv) A root system is called indecomposable or irreducible if it cannot bewritten as the direct sum of root systems of lower rank.
Proposition 11.3.4. Let Ψ = (X,R,X∨, R∨) be a semisimple root datum.Then R is a root system in the Q-vector space X ⊗Z Q.
Proof. Notice that 0 6∈ R because 〈α, α∨〉 = 2 for all α ∈ R. Moreover, for allα, β ∈ R, we have 〈β, α∨〉 ∈ Z because α∨ ∈ X∨, which shows that sα(β)−βis an integer multiple of α. The fact that Ψ is a semisimple root systemimplies that R spans V as a vector space. The other facts are clear. �
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Remark 11.3.5. As we already pointed out, the converse is more delicate,and there is no unique root datum that one can associate to a given rootsystem. Instead, the following is true: if (V,R) is a root system (where V isa finite-dimensional Q-vector space), then for any choice of a lattice X in Vlying between the root lattice P and the weight lattice Q of (V,R), there isa unique semisimple root datum Ψ = (X,R,X∨, R∨) w.r.t. the root system(V,R) and this choice of X. Since P has finite index in Q, there is onlya finite number of choices for X and hence a finite number of root dataassociated to the given root system (V,R).
We have seen in Theorem 11.2.11 that the root datum of a reductivelinear algebraic group over an algebraically closed field uniquely determinesthe group up to isomorphism. The root system does not determine the groupuniquely, but it almost does.
Theorem 11.3.6. Let G and G′ be two reductive linear algebraic groups overthe same algebraically closed field k, and let T and T ′ be maximal tori in Gand G′, respectively. Assume that Ψ(G, T ) and Ψ(G′, T ′) are root data withisomorphic root systems. Then G and G′ are isogenous. More precisely, thereis an isogeny from G to G′ mapping T to T ′.
Proof omitted. �
The root systems have been classified, and this will associate a certain typeto each reductive linear algebraic group. In order to describe the differentroot systems, we will need the notion of a base.
Definition 11.3.7. Let R be a root system in the Q-vector space V . Asubset S ⊂ R is called a base for R if it is a basis for V , and if each rootβ ∈ R can be written as β =
∑α∈Smα α, where the mα are integers of the
same sign, i.e. either all mα ≥ 0 or all mα ≤ 0. Once a base has been fixed,we refer to the elements of the base as the simple roots.
It is a non-trivial fact that every root system has a base, but we will takethis for granted.
Example 11.3.8. Consider the root system associated to G = GLn as inExample 11.2.4(4), i.e.
R = {ei − ej | 1 ≤ i, j ≤ n, i 6= j}.
ThenS = {e1 − e2, e2 − e3, . . . , en−1 − en}
is a base for R. (Recall that V is the Q-space spanned by R, which is thehyperplane x1 + x2 + · · ·+ xn = 0 inside Qn.)
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The following fact greatly reduces the possibilities for a root system.
Proposition 11.3.9. Let R be a root system in the Q-vector space V , andlet S be a base for R. Then:
(i) For any two distinct α, β ∈ S, the angle between α and β is either π/2,2π/3, 3π/4 or 5π/6. Moreover,
(a) If ∠(α, β) = 2π/3, then |α| = |β|;(b) If ∠(α, β) = 3π/4, then |α|/|β| =
√2 or 1/
√2;
(c) If ∠(α, β) = 5π/6, then |α|/|β| =√
3 or 1/√
3.
(ii) The root system is completely determined, up to isomorphism, by theset S of simple roots, the length of each of the simple roots, and theangles between any two distinct simple roots.
Proof omitted. �
The previous proposition will allow us to associate a diagram to each rootsystem which encodes the necessary information to recover the root systemcompletely.
Definition 11.3.10. Let R be a root system in the Q-vector space V , andlet S be a base for R. The Dynkin diagram of R is a graph with vertex set S,and where the edges can be either single, double or triple edges, dependingon the following rule:
(a) When ∠(α, β) = π/2, there is no edge between α and β;
(b) When ∠(α, β) = 2π/3, there is a single edge between α and β;
(c) When ∠(α, β) = 3π/4, there is a double edge between α and β;
(d) When ∠(α, β) = 5π/6, there is a triple edge between α and β.
Moreover, for each double or triple edge, we put an arrow pointing from thelongest root towards the shortest root.
Remark 11.3.11. The Dynkin diagram of a root system R is a connectedgraph if and only if R is an irreducible root system.
It turns out that there are exactly four different reduced root systems ofrank 2.
Proposition 11.3.12. Let R be a root system of rank 2. Then R is one ofthe following:
134
A1 × A1 Dynkin diagram
A2 Dynkin diagram
B2 Dynkin diagram
G2 Dynkin diagram
Proof omitted. �
We will now describe all irreducible root systems.
Theorem 11.3.13. Let R be an irreducible root system of arbitrary rank.
135
Then the Dynkin diagram of R is exactly one of the following.
An
Bn
Cn
Dn
E6, E7, E8
F4
G2
Proof omitted. �
We can now reformulate our main result.
Theorem 11.3.14. Let G be an almost simple linear algebraic group overan algebraically closed field k. Then up to isogeny, G is uniquely determinedby the field k and the root system R of G (w.r.t. an arbitrary maximal torusin G). Its type is one of An, Bn, Cn, Dn, E6, E7, E8, F4 or G2.
Proof. Since G is almost simple, the root datum of G is semisimple, re-duced, and irreducible. The result now follows from Theorem 11.3.6 andTheorem 11.3.13. �
136
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